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Chapter 1: Problem 93

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

Short Answer

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

Step by step solution

01

Finding inverse function of \(f(x)\)

The given function is \(f(x) = x + 4\). To find its inverse, swap \(x\) and \(y\) to get \(x = y + 4\), then solve for \(y\) to derive the inverse function. Hence, the inverse function becomes \(f^{-1}(x) = x - 4\).
02

Finding inverse function of \(g(x)\)

The given function is \(g(x) = 2x - 5\). To find its inverse, swap \(x\) and \(y\) to get \(x = 2y -5\), then solve for \(y\) to derive the inverse function. Hence, the inverse function becomes \(g^{-1}(x) = (x + 5)/2\).
03

Composing inverse functions

Now compose the functions \(g^{-1}\) and \(f^{-1}\) i.e. \(g^{-1}\circ f^{-1} = g^{-1}(f^{-1}(x))\). Substituting the derived inverse functions, it becomes \(g^{-1}(f^{-1}(x)) = (f^{-1}(x) + 5) / 2\). On simplifying this further by substituting \(f^{-1}(x)\) from step 1, it becomes \((x - 4 + 5) / 2\), simplifying to finally \(g^{-1}\circ f^{-1} = (x + 1) / 2\).

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Most popular questions from this chapter

Property tax is based on the assessed value of a property. A house that has an assessed value of \(\$ 150,000\) has a property tax of \(\$ 5520\). Find a mathematical model that gives the amount of property \(\operatorname{tax} y\) in terms of the assessed value \(x\) of the property. Use the model to find the property tax on a house that has an assessed value of \(\$ 225,000\).

(a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\frac{x+1}{x-2} $$

(a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\frac{4}{x} $$

(a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\frac{6 x+4}{4 x+5} $$

Find a mathematical model for the verbal statement. For a constant temperature, the pressure \(P\) of a gas is inversely proportional to the volume \(V\) of the gas.
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