Problem 77
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). $$f(x)=\sqrt[3]{2-x}$$
Step-by-Step Solution
Verified Answer
Yes, the function \(f(x)=\sqrt[3]{2-x}\) has an inverse that is also a function, meaning the function is one-to-one.
1Step 1: Plotting the function
Start by plotting the function \(f(x)=\sqrt[3]{2-x}\) using a graphing utility. This type of function will typically resemble a curve due to the cubic root. Since \(f(x)\) is defined everywhere, the domain is all real numbers.
2Step 2: Checking for one-to-oneness using Horizontal Line Test
Apply the Horizontal Line Test to the graph. If no horizontal line intersects the graph more than once, then the function is one-to-one. Otherwise, if any horizontal line intersects the graph more than once, the function is not a one-to-one function.
3Step 3: Conclusion
After doing the Horizontal Line Test, you will find that no horizontal line intersects the graph of the function more than once. Hence, \(f(x) = \sqrt[3]{2-x}\) is a one-to-one function and has an inverse that is also a function.
Other exercises in this chapter
Problem 76
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 77
What is a circle? Without using variables, describe how the definition of a circle can be used to obtain a form of its equation.
View solution Problem 77
Express the given function \(h\) as a composition of two functions \(f\) and \(g\) so that \(h(x)=(f \circ g)(x)\). $$h(x)=\sqrt[3]{x^{2}-9}$$
View solution Problem 77
Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$h(x)=\sqrt{x+2}-2$$
View solution