Using only a graphing calculator, determine whether the...

Using only a graphing calculator, determine whether the functions are inverses of each other. Consider the function $f$ given by $$ f(x)=\left\{\begin{array}{ll} x^{3}+2, & x \leq-1 \\ x^{2}, & -1<x<1 \\ x+1, & x \geq 1 \end{array}\right. $$ Does $f$ have an inverse that is a function? Why or why not?

Using only a graphing calculator, determine whether the functions are inverses of each other.
Consider the function $f$ given by
$$
f(x)=\left\{\begin{array}{ll}
x^{3}+2, & x \leq-1 \\
x^{2}, & -1<x<1 \\
x+1, & x \geq 1
\end{array}\right.
$$
Does $f$ have an inverse that is a function? Why or why not?

Key Concepts

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain. When assessing a piecewise function for invertibility, it is essential to examine each segment to ensure the entire function is one-to-one. If any segment fails the horizontal line test, the overall function will not have an inverse that is a function.

Horizontal Line Test

The horizontal line test is a graphical method used to determine if a function is one-to-one. If any horizontal line intersects the graph of the function at more than one point, this indicates that the function does not have a unique output for each input value, and hence, its inverse would not be a function.

Inverse Functions

Inverse functions reverse the operation of the original function. For a function f to have an inverse, often denoted as f?¹, each output of f must correspond to exactly one input. This property ensures that the process of 'undoing' the function is well-defined and that when composed with f, the inverse returns the original value.

One-to-One Functions

A function is one-to-one if each value in the range is produced by exactly one value from the domain. This unique mapping is crucial because it guarantees that no output is repeated for different inputs, ensuring that the inverse relation is also a function. Without this property, the inverse relation may fail to be a function.

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