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Chapter 1: Problem 71

Determine whether the function has an inverse function. If it does, then find the inverse function. $$f(x)=\frac{6 x+4}{4 x+5}$$

Short Answer

Expert verified
Yes, the function \(f(x)=\frac{6 x+4}{4 x+5}\) does have an inverse function. The inverse function is given by \(f^{-1}(x)=\frac{4-5x}{4x-6}\).

Step by step solution

01

Check if the equation has an inverse

We can find out if there is an inverse by checking if the function is one-to-one, meaning each x-value maps to a unique y-value and vice versa 鈥� a necessary condition for a function to have an inverse. In this case, it's not easily checkable due to function's complex nature. This does not mean there isn鈥檛 one 鈥� just that we must proceed with our calculations and not rely on this test in this instance.
02

Find the inverse

To find the inverse, we replace \(f(x)\) with \(y\), so our equation becomes \(y=\frac{6x+4}{4x+5}\). To find the inverse from this, we first interchange x and y and solve the equation for y, thus finding \(f^{-1}(x)\). Thus, we have: \(x=\frac{6y+4}{4y+5}\). Next, we will solve this equation for y.
03

Solve the equation for y

First, cross-multiply to get rid of the fraction: \(x(4y+5)=6y+4\). Distribute x on the left side to get: \(4xy+5x=6y+4\). Rearrange the equation, grouping the y terms on one side and constants on the other side: \(4xy-6y=4-5x\). Factor out the y: \(y(4x-6)=4-5x\). Finally, divide by \(4x-6\) on both sides to solve for y: \(y=\frac{4-5x}{4x-6}\).
04

Confirm the inverse

To confirm the inverse works, you can substitute a value into the inverse function, then substitute the result into the original function, and you should end up with the initial substitution. If you do not, there has likely been a mistake in the calculation.

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Most popular questions from this chapter

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$f(x)=|x-1|$$

An oceanographer took readings of the water temperatures \(C\) (in degrees Celsius) at several depths \(d\) (in meters). The data collected are shown as ordered pairs \((d, C)\) (Spreadsheet at LarsonPrecalculus.com) $$\begin{aligned} &(1000,4.2) \quad(4000,1.2)\\\ &(2000,1.9) \quad(5000,0.9)\\\ &(3000,1.4) \end{aligned}$$ A.Sketch a scatter plot of the data. B. Does it appear that the data can be modeled by the inverse variation model \(C=k / d ?\) If so, find \(k\) for each pair of coordinates. C. Determine the mean value of \(k\) from part (b) to find the inverse variation model \(C=k / d\) D. Use a graphing utility to plot the data points and the inverse model from part (c). E. Use the model to approximate the depth at which the water temperature is \(3^{\circ} \mathrm{C}\)

Determine whether the statements use the word function in ways that are mathematically correct. Explain your reasoning. (a) The amount in your savings account is a function of your salary. (b) The speed at which a free-falling baseball strikes the ground is a function of the height from which it was dropped.

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