Question

3. Look at the graph of a function below. Will the inverse of this function be a function as well? If so, state the domain and range of the inverse function. If not, make a restriction on the domain of the inverse so that the inverse is a function. 2- 1- 0 2​

Answer 1

• inverse is not a function
• unless the domain is restricted to |x| ≥ 1.2 (approximately)

Explanation:

The test to see if the inverse function is also a function is called the "horizontal line test." The test passes if any horizontal line intersects the graph in only one place.

Here, a horizontal line can intersect the graph in 1, 2, or 3 places, so the test fails. The function does not have an inverse that is a function.

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If the domain of the inverse relation is restricted to |x| > 1.2, then that inverse will map any x to only a single value of y. Then it will be a function.

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The graph shows the original function (dashed red line) and the inverse relation (blue). The green shading marks values of x for which there is a single value of y, so the inverse relation is a function in those regions.

(We could be more specific as to the limits on the domain of f^-1(x), but the given graph seems to have an unknown vertical scale factor.)