Inverses Represented Graphically Quick Check

Question

4 of 54 of 5 Items

Question
Use the images to answer the question.

A coordinate plane's x-axis ranges from negative 9 to 9 and its y-axis ranges from negative 11 to 11, both by 1-unit increments. A curve is plotted labeled f of x equals x cubed and is plotted passing through 5 marked points.

The function f left parenthesis x right parenthesis equals the cube root of x is graphed on a coordinate plane. Five points are plotted and labeled along the length of the curve.

Determine if the cubic function and the cube root function are inverses.

(1 point)
Responses

Yes, all corresponding ordered pairs are reflections of one another across the line y=−x
.
Yes, all corresponding ordered pairs are reflections of one another across the line y equals negative x .

Yes, all corresponding ordered pairs are reflections of one another across the line y=x
.
Yes, all corresponding ordered pairs are reflections of one another across the line y equals x .

No, not all corresponding ordered pairs are reflections of one another across the line y=x
.
No, not all corresponding ordered pairs are reflections of one another across the line y equals x .

No, not all corresponding ordered pairs are reflections of one another across the line y=−x
.
No, not all corresponding ordered pairs are reflections of one another across the line y equals negative x .
Skip to navigation

Answer 1

To determine if the cubic function \( f(x) = x^3 \) and the cube root function \( f(x) = \sqrt[3]{x} \) are inverses of each other, we need to check the reflection properties across the line \( y = x \).

1. The function \( f(x) = x^3 \) takes an input \( x \) and gives an output \( x^3 \).
2. The inverse function \( f^{-1}(x) = \sqrt[3]{x} \) takes an input \( x \) and returns \( x \) raised to the power of \( \frac{1}{3} \).

For two functions to be inverses of each other, the following relationship holds:

• If \( f(a) = b \), then \( f^{-1}(b) = a \) for all values \( a \) in the domain of \( f \).

Graphically, this means that the points of \( f(x) \) and \( f^{-1}(x) \) must reflect across the line \( y = x \).

Thus, if you plot both functions on the coordinate plane, the points (a, b) from the cubic function should be reflected to the points (b, a) from the cube root function. If they meet this criterion, they are indeed inverses.

The correct response to your question is:

Yes, all corresponding ordered pairs are reflections of one another across the line \( y = x \).