Problem 76
Question
Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$f(x)=x^{2}-1$$
Step-by-Step Solution
Verified Answer
Based on the graph and the horizontal line test, the function \(f(x) = x^2 - 1\) does not have an inverse that is a function, because it is not one-to-one.
1Step 1: Graph the Function
First, graph the function \(f(x) = x^2 - 1\) using a graphing utility. This function is a parabola shifted one unit down from the origin, opening upward.
2Step 2: Assess the Graph
Next, make an assessment of the generated graph. Deduce whether or not a horizontal line drawn through the graph crosses the plot more than once. For the function \(f(x) = x^2 - 1\), a horizontal line will intersect the graph at more than one point.
3Step 3: Conclusion
Based on the horizontal line test, the function \(f(x) = x^2 - 1\) is not one-to-one because any horizontal line above or equal to -1 intersects the graph at more than one point. Therefore, the function does not have an inverse that is also a function.
Other exercises in this chapter
Problem 75
Find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers.
View solution Problem 76
In your own words, describe how to find the midpoint of a line segment if its endpoints are known.
View solution Problem 76
find and simplify the difference quotient $$ \frac{f(x+h)-f(x)}{h}, h \neq 0 $$ for the given function. $$ f(x)=\sqrt{x-1} $$
View solution Problem 76
Express the given function \(h\) as a composition of two functions \(f\) and \(g\) so that \(h(x)=(f \circ g)(x)\). $$h(x)=(2 x-5)^{3}$$
View solution