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Chapter 1: Problem 111

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

Short Answer

Expert verified
The composition of the inverse functions \(g^{-1} \circ f^{-1}(x)\) is \((x+1)/2\).

Step by step solution

01

Find the Inverse of Function f

To find the inverse of function f, denoted by \(f^{-1}(x)\), we interchange \(f(x)\) with \(x\) and solve for \(x\). In this case, \(f(x) = x + 4\) becomes \(f^{-1}(x) = x - 4\).
02

Find the Inverse of Function g

To obtain \(g^{-1}(x)\), we repeat a similar process. We interchange \(g(x)\) with \(x\) and solve for \(x\). Given \(g(x) = 2x - 5\), the inverse function is determined as follows: \(g^{-1}(x) = (x + 5)/2\).
03

Determine the Composition

Now, to obtain \(g^{-1} \circ f^{-1}(x)\), we have to substitute \(f^{-1}(x)\) into \(g^{-1}(x)\). Plugging \(x - 4\) from \(f^{-1}(x)\) into \(g^{-1}(x) = (x + 5)/2\), we get \(g^{-1} \circ f^{-1}(x) = ((x - 4) + 5)/2 = (x + 1)/2\).

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Most popular questions from this chapter

Proof Prove that if \(f\) and \(g\) are one-to-one functions, then \((f \circ g)^{-1}(x)=\left(g^{-1} \circ f^{-1}\right)(x)\).

Determine whether the statement is true or false. Justify your answer. It is possible for an odd function to have the interval \([0, \infty)\) as its domain.

(a) use a graphing utility to graph the function \(f,\) (b) use the draw inverse feature of the graphing utility to draw the inverse relation of the function, and (c) determine whether the inverse relation is an inverse function. Explain your reasoning. $$f(x)=\frac{3 x^{2}}{x^{2}+1}$$

Find three points that lie on the graph of the equation. (There are many correct answers.) $$x^{2}+y^{2}=49$$

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 \(x\). $$f(x)=x \sqrt{1-x^{2}}$$
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