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Chapter 1: Problem 31

Find the inverse function (on the given interval, if specified) and graph both fand \(f^{-1}\) on the same set of axes. Check your work by looking for the required symmetry in the graphs. $$f(x)=8-4 x$$

Short Answer

Expert verified
Answer: The inverse function of \(f(x)=8-4x\) is \(f^{-1}(x)=2-\frac{1}{4}x\). When graphing both functions on the same set of axes, we can confirm their correctness by observing the symmetry between the functions around the line y=x. If the points of the original function are reflected over the line y=x and they line up with the points of the inverse function, we have confirmed the correctness of the inverse function.

Step by step solution

01

Find the inverse function \(f^{-1}(x)\)

To find the inverse function, first replace f(x) with y: $$y=8-4x$$ Now, swap the roles of x and y: $$x=8-4y$$ Solve for y to obtain the inverse function: $$4y=8-x$$ $$y=\frac{1}{4}(8-x)=2-\frac{1}{4}x$$ So, the inverse function is: $$f^{-1}(x)=2-\frac{1}{4}x$$
02

Graph both functions on the same set of axes

To graph both functions, choose a set of x values and corresponding y-values for both the original function and its inverse. For example: | \(x\) | \(f(x)\) | \(f^{-1}(x)\) | |-----|-------|-------------| | -2 | 16 | 3 | | -1 | 12 | 2.25 | | 0 | 8 | 2 | | 1 | 4 | 1.75 | | 2 | 0 | 1 | Now plot these points on a single set of axes and connect the points for the graphs of both the original function (in blue) and its inverse (in red).
03

Check for required symmetry between the graphs of the original function and its inverse

The required symmetry between a function and its inverse is around the line y=x. In other words, if you can reflect the points of the original function over the line y=x and it lines up with the points of the inverse function, then you have found the correct inverse function. After plotting the two functions on the same set of axes, we observe that they have symmetry around the line y=x, which confirms that we have found the correct inverse function and graphed them correctly.

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