🚧 Setup for 7.3
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@ -3330,3 +3330,182 @@ Omitted.
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compute values of the function.
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Omitted.
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---
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Page 494
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**Exercise Set 7.3**
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In each of 1 and 2, functions $f$ and $g$ are defined by arrow diagrams. Find
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$g \circ f$ and $f \circ g$ and determine whether $g \circ f$ equals
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$f \circ g$.
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1. (See page 494 for image)
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2. (See page 494 for image)
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In 3 and 4, functions $F$ and $G$ are defined by formulas. Find $G \circ F$ and
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$F \circ G$ and determine whether $G \circ F$ equals $F \circ G$.
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3. $F(x) = x^3$ and $G(x) = x - 1$, for each real number $x$.
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4. $F(x) = x^5$ and $G(x) = x^{\frac{1}{5}}$ for each real number $x$.
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5. Define $f: \mathbb{R} \to \mathbb{R}$ by the rule $f(x) = -x$ for every real
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number $x$. Find $(f \circ f)(x)$.
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6. Define $F: \mathbb{Z} \to \mathbb{Z}$ and $G: \mathbb{Z} \to \mathbb{Z}$ by
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the rules $F(a) = 7a$ and $G(a) = a \mod 5$ for each integer $a$. Find
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$(G \circ F)(0)$, $(G \circ F)(1)$, $(G \circ F)(2)$, $(G \circ F)(3)$, and
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$(G \circ F)(4)$.
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7. Define $L: \mathbb{Z} \to \mathbb{Z}$ and $M: \mathbb{Z} \to \mathbb{Z}$ by
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the rules $L(a) = a^2$ and $M(a) = a \mod 5$ for each integer $a$.
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a. Find $(L \circ M)(12)$, $(M \circ L)(12)$, $(L \circ M)(9)$, and
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$(M \circ L)(9)$.
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b. Is $L \circ M = M \circ L$?
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8. Let $S$ be the set of all strings in _a_'s and _b_'s and let
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$L: S \to \mathbb{Z}$ be the length function:
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For all strings $s \in S$ ,
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$$ L(s) = \text{ the number of characters in } s $$
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Let $T: \mathbb{Z} \to \{0, 1, 2\}$ be the $\mod 3$ function:
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$$ \text{For every integer } n, \quad T(n) = n \mod 3 $$
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a. $(T \circ L)(abaa) = \text{ ?}$
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b. $(T \circ L)(baaab) = \text{ ?}$
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c. $(T \circ L)(aaa) = \text{ ?}$
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9. Define $F: \mathbb{R} \to \mathbb{R}$ and $G: \mathbb{R} \to \mathbb{Z}$ by
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the following formulas: $F(x) = \dfrac{x^2}{3}$ and
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$G(x) = \lfloor x \rfloor$ for every $x \in \mathbb{R}$.
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a. $(G \circ F)(2) = \text{ ?}$
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b. $(G \circ F)(-3) = \text{ ?}$
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c. $(G \circ F)(5) = \text{ ?}$
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10. Define $F: \mathbb{Z} \to \mathbb{Z}$ and $G: \mathbb{Z} \to \mathbb{Z}$ by
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the rules $F(n) = 2n$ and $G(n) = \left\lfloor \dfrac{n}{2} \right\rfloor$
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for every integer $n$.
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a. Find $(G \circ F)(8)$, $(F \circ G)(8)$, $(G \circ F)(3)$, and
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$(F \circ G)(3)$.
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b. Is $G \circ F = F \circ G$? Explain.
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11. Define $F: \mathbb{R} \to \mathbb{R}$ and $G : \mathbb{R} \to \mathbb{R}$ by
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the rules $F(n) = 3x$ and $G(n) = \left\lceil \dfrac{x}{3} \right\rceil$ for
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every real number $x$.
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a. Find $(G \circ F)(6)$, $(F \circ G)(6)$, $(G \circ F)(1)$, and
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$(F \circ G)(1)$.
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b. Is $G \circ F = F \circ G$? Explain.
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The functions of each pair in 12-14 are inverse to each other. For each pair,
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check that both compositions give the identity function.
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12. $F: \mathbb{R} \to \mathbb{R}$ and $F^{-1}: \mathbb{R} \to \mathbb{R}$ are
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defined by
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$$ F(x) = 3x + 2 \quad \text{ and } \quad F^{-1}(y) = \frac{y - 2}{3} $$
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for every $y \in \mathbb{R}$.
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13. $G: \mathbb{R}^+ \to \mathbb{R}^+$ and
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$G^{-1}: \mathbb{R}^+ \to \mathbb{R}^+$ are defined by
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$$ G(x) = x^2 \quad \text{ and } \quad G^{-1}(x) = \sqrt{x} $$
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for every $x \in \mathbb{R}^+$.
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14. $H$ and $H^{-1}$ are both defined from $\mathbb{R} - \{1\}$ to
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$\mathbb{R} - \{1\}$ by the formula
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$$ H(x) = H^{-1}(x) = \frac{x + 1}{x - 1}, \quad \text{ for each } x \in \mathbb{R} - \{1\} $$
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15. Explain how it follows from the definition of logarithm that
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a. $\log_{b}(b^x) = x$, for every real number $x$.
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b. $b^{\log_{b}x} = x$, for every positive real number $x$.
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16. Prove Theorem 7.3.1(b): If $f$ is any function from a set $X$ to a set $Y$,
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then $I_y \circ f = f$, where $I_y$ is the identity function on $Y$.
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17. Prove Theorem 7.3.2(b): If $f: X \to Y$ is a one-to-one and onto function
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with inverse function $f^{-1}: Y \to X$, then $f \circ f^{-1} = I_y$, where
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$I_y$ is the identity function on $Y$.
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18. Suppose $Y$ and $Z$ are sets and $g: Y \to Z$ is a one-to-one function. This
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means that if $g$ takes the same value on any two elements of $Y$, then
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those elements are equal. Thus, for example, if $a$ and $b$ are elements of
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$Y$ and $g(a) = g(b)$, then it can be inferred that $a = b$. What can be
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inferred in the following situations?
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a. $s_k$ and $s_m$ are elements of $Y$ and $g(s_k) = g(s_m)$.
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b. $\dfrac{z}{2}$ and $\dfrac{t}{2}$ are elements of $Y$ and
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$g\left(\dfrac{z}{2}\right) = g\left(\dfrac{t}{2}\right)$.
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c. $f(x_1)$ and $f(x_2)$ are elements of $Y$ and $g(f(x_1)) = g(f(x_2))$.
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19. If $f: X \to Y$ and $g: Y \to Z$ are functions and $g \circ f$ is
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one-to-one, must $g$ be one-to-one? Prove or give a counterexample.
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20. If $f: X \to Y$ and $g: Y \to Z$ are functions and $g \circ f$ is onto, must
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$f$ be onto? Prove or give a counterexample.
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21. If $f: X \to Y$ and $g: Y \to Z$ are functions and $g \circ f$ is
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one-to-one, must $f$ be one? Prove or give a counterexample.
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22. If $f: X \to Y$ and $g: Y \to Z$ are functions and $g \circ f$ is onto, must
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$g$ be onto? Prove or give a counterexample.
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23. Let $f: W \to X$, $g: X \to Y$, and $h: Y \to Z$ be functions. Must
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$h \circ (g \circ f) = (h \circ g) \circ f$? Prove or give a counterexample.
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24. True or False? Given any set $X$ and given any functions $f: X \to X$,
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$g: X \to X$, and $h: X \to X$, if $h$ is one-to-one and
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$h \circ f = h \circ g$, then $f = g$. Justify your answer.
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25. True or False? Given any set $X$ and given any functions $f: X \to X$,
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$g: X \to X$, and $h: X \to X$, if $h$ is one-to-one and
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$f \circ h = g \circ h$, then $f = g$. Justify your answer.
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In 26 and 27 find $(g \circ f)^{-1}$, $g^{-1}$, $f^{-1}$, and
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$f^{-1} \circ g^{-1}$, and state how $(g \circ f)^{-1}$ and
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$f^{-1} \circ g^{-1}$ are related.
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26. Let $X = \{a, b, c\}$, $Y = \{x, y, z\}$, and $Z = \{u, v, w\}$. Define
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$f: X \to Y$ and $g: Y \to Z$ by the arrow diagrams below.
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(See page 495 for image.)
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27. Define $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ by
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the formulas
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$$ f(x) = x + 3 \quad \text{ and } \quad g(x) = -x \quad \text{ for each } x \in \mathbb{R} $$
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28. Prove or give a counterexample: If $f: X \to Y$ and $g: Y \to X$ are
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functions such that $g \circ f = I_x$ and $f \circ g = I_y$, then $f$ and
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$g$ are both one-to-one and onto and $g = f^{-1}$.
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29. Suppose $f: X \to Y$ and $g: Y \to Z$ are both one-to-one and onto. Prove
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that $(g \circ f)^{-1}$ exists and that
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$(g \circ f)^{-1} = f^{-1} \circ g^{-1}$.
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30. Let $f: X \to Y$ and $g: Y \to Z$. Is the following property true or false?
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For every subset $C$ in $Z$, $(g \circ f)^{-1}(C) = f^{-1}(g^{-1}(C))$.
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Justify your answer.
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@ -258,3 +258,158 @@ be shown.]_
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Suppose $x \in X$. _[We must show that there exists an element $y$ in $Y$ such
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that $F^{-1}(y) = x$.]_ Let $y = F(x)$. Then $y \in Y$, and by definition of
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$F^{-1}$, $F^{-1}(y) = x$ _[as was to be shown.]_
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---
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Page 485
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**Definition**
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Let $f: X \to Y$ and $g: Y' \to Z$ be functions with the property that the range
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of $f$ is a subset of the domain of $g$. Define a new function
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$g \circ f: X \to Z$ as follows:
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$$ (g \circ f)(x) = g(f(x)) \quad \text{ for each } x \in X $$
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where $g \circ f$ is read "$g$ circle $f$" and $g(f(x))$ is read "$g$ of $f$ of
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$x$." The function $g \circ f$ is called the **composition of $f$ and $g$**.
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---
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Page 487
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**Theorem 7.3.1 Composition with an Identity Function**
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If $f$ is a function from a set $X$ to a set $Y$, and $I_x$ is the identity
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function on $X$, and $I_y$ is the identity function on $Y$, then
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$$ \text{(a) } f \circ I_x = f \quad \text{ and } \quad \text{(b) } I_y \circ f = f $$
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**Proof:**
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_Part (a):_
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Suppose $f$ is a function from a set $X$ to a set $Y$ and $I_x$ is the identity
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function on $X$. Then, for each $x$ in $X$,
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$$ (f \circ I_x)(x) = f(I_x(x)) = f(x) $$
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Hence, by the definition of equality of functions, $f \circ I_x = f$, as was to
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be shown.
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_Part (b):_
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This is exercise 16 at the end of this section.
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---
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Page 488
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**Theorem 7.3.2 Composition of a Function with Its Inverse**
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If $f: X \to Y$ is a one-to-one and onto function with inverse function
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$f^{-1}: Y \to X$, then
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$$ \text{(a) } f^{-1} \circ f = I_x \quad \text{ and } \quad \text{(b) } f \circ f^{-1} = I_y $$
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**Proof:**
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_Part (a):_
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Suppose $f: X \to Y$ is a one-to-one and onto function with inverse function
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$f^{-1}: Y \to X$. _[To show that $f^{-1} \circ f = I_x$, we must show that for
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each $x \in X$, $(f^{-1} \circ f)(x) = x$.]_ Let $x$ be any element in $X$.
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Then, by definition of composition of functions,
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$$ (f^{-1} \circ f)(x) = f^{-1}(f(x)) $$
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Let
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$$ z = f^{-1}(f(x)) $$
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By the definition of inverse function,
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$$ f(z) = f(x) $$
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and, because $f$ is one-to-one, this implies that
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$$ z = x $$
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Now $z = f^{-1}(f(x))$ also, and so, by substitution,
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$$ f^{-1}(f(x)) = x $$
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Or, equivalently,
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$$ (f^{-1} \circ f)(x) = x $$
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_[as was to be shown]._
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Since $x$ is any element of $X$ and since $I_x(x) = x$, this proves that
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$f^{-1} \circ f = I_x$.
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_Part (b):_
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This is exercise 17 at the end of this section.
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---
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Page 490
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**Theorem 7.3.3**
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If $f: X \to Y$ and $g: Y \to Z$ are both one-to-one functions, then $g \circ f$
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is one-to-one.
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---
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Page 491
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**Proof of Theorem 7.3.3:**
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Suppose $f: X \to Y$ and $g: Y \to Z$ are both one-to-one functions. _[We must
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show that $g \circ f$ is one-to-one.]_ Suppose $x_1$ and $x_2$ are elements of
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$X$ such that
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$$ (g \circ f)(x_1) = (g \circ f)(x_2) $$
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_[We must show that $x_1 = x_2$.]_ By definition of composition of functions,
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$$ g(f(x_1)) = g(f(x_2)) $$
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Since $g$ is one-to-one,
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$$ f(x_1) = f(x_2) $$
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And since $f$ is one-to-one,
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$$ x_1 = x_2 $$
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_[as was to be shown]._ Hence $g \circ f$ is one-to-one.
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---
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Page 491
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**Theorem 7.3.4**
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If $f: X \to Y$ and $g: Y \to Z$ are both onto functions, then $g \circ f$ is
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onto.
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---
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Page 493
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**Proof of Theorem 7.3.4**
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Suppose $f: X \to Y$ and $g: Y \to Z$ are both onto functions. _[We must show
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that $g \circ f$ is onto.]_ Let $z$ be any _[particular but arbitrarily chosen]_
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element of $Z$. _[We must show the existence of an element in $X$ such that
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$g \circ f$ of that element equals $z$.]_ Since $g$ is onto, there is an
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element, say $y$, in $Y$ such that $g(y) = z$. And since $f$ is onto, there is
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an element, say $x$, in $X$ such that $f(x) = y$. Hence there is an element $x$
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in $X$ such that
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$$ (g \circ f)(x) = g(f(x)) = g(y) = z $$
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_[as was to be shown]._ It follows that $g \circ f$ is onto.
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@ -116,3 +116,28 @@ function from $X$ to $Y$; both one-to-one and onto
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the unique element $x$ in $X$ such that $F(x) = y$ (in other words, $F^{-1}(y)$
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is the unique preimage of $y$ in $X$)
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---
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Page 494
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**Test Yourself**
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1. If $f$ is a function from $X$ to $Y'$, $g$ is a function from $Y \to Z$, and
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$Y' \subseteq Y$, then $g \circ f$ is a function from _____ to _____, and
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$(g \circ f)(x) =$ _____ for every $x$ in $X$.
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2. If $f$ is a function from $X$ to $Y$ and $I_x$ and $I_y$ are the identity
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functions from $X$ to $X$ and $Y$ to $Y$, respectively, then $f \circ I_x =$
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_____ and $I_y \circ f =$ _____.
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3. If $f$ is a one-to-one correspondence from $X$ to $Y$, then
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$f^{-1} \circ f =$ _____ and $f \circ f^{-1} =$ _____.
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4. If $f$ is a one-to-one function from $X$ to $Y$ and $g$ is a one-to-one
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function from $Y$ to $Z$, you prove that $g \circ f is one-to-one by
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supposing that _____ and then showing that _____.
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5. If $f$ is an onto function from $X$ to $Y$ and $g$ is an onto function from
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$Y$ to $Z$, you prove that $g \circ f$ is onto by supposing that _____ and
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then showing that _____.
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