P.4 Assess Your Understanding

Concepts and Vocabulary

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True or False  If every vertical line intersects the graph of a function \(f\) at no more than one point, \(f\) is a one-to-one function.

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If the domain of a one-to-one function \(f\) is \([4,\infty)\), the range of its inverse function \(f^{-1}\) is _____.

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True or False  If \(f\) and \(g\) are inverse functions, the domain of \(f\) is the same as the domain of \(g\).

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True or False  If \(f\) and \(g\) are inverse functions, their graphs are symmetric with respect to the line \(y = x\).

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True or False  If \(f\) and \(g\) are inverse functions, then \((f \circ g) (x) = f(x) \,\cdot\, g(x)\).

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True or False  If a function \(f\) is one-to-one, then \(f (f^{-1}( x)) = x\), where \(x\) is in the domain of \(f\).

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Given a collection of points \((x,y)\), explain how you would determine if it represents a one-to-one function \(y = f(x)\).

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Given the graph of a one-to-one function \(y = f(x)\), explain how you would graph the inverse function \(f^{-1}\).

Practice Problems

In Problems 9–14, the graph of a function f is given. Use the Horizontal-line Test to determine whether f is one-to one.

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In Problems 15–18, verify that the functions f and g are inverses of each other by showing that \((f\circ g)(x) = x\) and \((g\circ f)(x) = x\).

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\(f( x) =3x+4\); \(g( x) = \dfrac{1}{3}( x-4)\)

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\(f( x) = x^{3}-8\); \(g(x) = \sqrt[3]{x+8}\)

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\(f(x) = \dfrac{1}{x}\); \(g(x) = \dfrac{1}{x}\)

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\(f(x) = \dfrac{2x+3}{x+4}\); \(g(x) = \dfrac{4x-3}{2-x}\)

In Problems 19–22, (a) determine whether the function is one-to-one. If it is one-to-one, (b) find the inverse of each one-to-one function. (c) State the domain and the range of the function and its inverse.

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\(\{(-3,5), (-2,9), (-1,2), (0,11), ( 1,-5)\}\)

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\(\{(-2,2), (-1,6), ( 0,8), ( 1,-3), ( 2,8)\}\)

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\(\{(-2,1), (-3,2), (-10,0), (1,9), ( 2,1)\}\)

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\(\{(-2,-8), (-1,-1), (0,0), (1,1), (2,8)\}\)

In Problems 23–28, the graph of a one-to-one function f is given. Draw the graph of the inverse function. For convenience, the graph of \(y=x\) is also given.

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In Problems 29–38, the function f is one-to-one.

  1. Find its inverse and check the result.
  2. Find the domain and the range of f and the domain and the range of \(f^{-1}\).

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\(f(x) = 4x + 2\)

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\(f(x) = 1-3x\)

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\(f(x) = \sqrt[3]{x+10}\)

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\(f(x) = 2x^{3}+4\)

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\(f(x) = \dfrac{1}{x-2}\)

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\(f(x) = \dfrac{2x}{3x-1}\)

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\(f(x) = \dfrac{2x+3}{x+2}\)

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\(f(x) = \dfrac{-3x-4}{x-2}\)

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\(f(x) = x^{2}+4\), \(x\geq 0\)

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\(f(x) = (x-2)^{2}+4\), \(x\leq 2\)