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Chapter 1: Problem 112

Use the functions \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$f^{-1} \circ g^{-1}$$

Short Answer

Expert verified
The function \(f^{-1} \circ g^{-1}\) equals \(x/2 - 2\).

Step by step solution

01

Find the inverse of function \(f(x)\)

The given \(f(x) = x+4\). To find the inverse, let \(y = f(x)\), which gives \(y = x + 4\). By rearranging this equation to solve for \(x\), we get \(x = y - 4\). Substituting \(f^{-1}(y)\) for \(x\), we get \(f^{-1}(y) = y - 4\).
02

Find the inverse of function \(g(x)\)

The given \(g(x) = 2x - 5\). To find the inverse, let \(y = g(x)\), which gives \(y = 2x - 5\). By rearranging this equation to solve for \(x\), we get \(x = (y + 5)/2\). Substituting \(g^{-1}(y)\) for \(x\), we get \(g^{-1}(y) = (y + 5)/2\).
03

Find the composition \(f^{-1} \circ g^{-1}\)

The notation \(f^{-1} \circ g^{-1}\) means that the function \(g^{-1}\) is evaluated first, and then the function \(f^{-1}\) is evaluated. That is, \(f^{-1}(g^{-1}(x)) = f^{-1}((x+5)/2) = (x+5)/2 - 4 = x/2 - 2\). So, the function \(f^{-1} \circ g^{-1}\) equals \(x/2 - 2\).

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Most popular questions from this chapter

Determine whether the function is even, odd, or neither (a) algebraically, (b) graphically by using a graphing utility to graph the function, and (c) numerically by using the table feature of the graphing utility to compare \(f(x)\) and \(f(-x)\) for several values of \(x\). $$f(t)=t^{2}+2 t-3$$

Use the functions \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$(g \circ f)^{-1}$$

Identify the terms. Then identify the coefficients of the variable terms of the expression. $$\frac{x}{3}-5 x^{2}+x^{3}$$

Determine whether the statement is true or false. Justify your answer.The graphs of \(f(x)=|x|+6\) and \(f(x)=|-x|+6\) are identical.

Determine whether the function is even, odd, or neither (a) algebraically, (b) graphically by using a graphing utility to graph the function, and (c) numerically by using the table feature of the graphing utility to compare \(f(x)\) and \(f(-x)\) for several values of \(x\). $$g(x)=x^{3}-5 x$$
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