Question

1. here is a function:

$f(x) = 3x^3 - 2$

a) write an equation for the inverse function. $f^{-1}(x) = \square$

b) here are some true statements about $f(x)$:

• $f(x)$ has a y-intercept at $(0, -2)$.
• $f(2)=22$.
• $f(x)$ is a cubic polynomial function.

write two true statements about the inverse function $f^{-1}(x)$.
\square

show your work

Explanation:

Part (a)

Step 1: Swap \( x \) and \( y \)

Let \( y = 3x^3 - 2 \). Swap \( x \) and \( y \) to get \( x = 3y^3 - 2 \).

Step 2: Solve for \( y \)

First, add 2 to both sides: \( x + 2 = 3y^3 \). Then divide both sides by 3: \( \frac{x + 2}{3} = y^3 \). Finally, take the cube root of both sides: \( y = \sqrt[3]{\frac{x + 2}{3}} \). So the inverse function \( f^{-1}(x) = \sqrt[3]{\frac{x + 2}{3}} \).

1. For the \( x \)-intercept of \( f(x) \) at \( (0, -2) \), the inverse function \( f^{-1}(x) \) will have a \( y \)-intercept at \( (-2, 0) \) (since the inverse function swaps the \( x \) and \( y \) coordinates of the original function's intercepts).
2. Given \( f(2) = 22 \), for the inverse function, we have \( f^{-1}(22) = 2 \) (because if \( f(a) = b \), then \( f^{-1}(b) = a \)). Also, since \( f(x) \) is a cubic polynomial (and thus one - to - one), \( f^{-1}(x) \) is also a function (specifically, a cube - root function, which is a type of odd - degree root function and is one - to - one). Another true statement: \( f^{-1}(x) \) is a one - to - one function (because the original function \( f(x) \) is a cubic function, which is strictly increasing (its derivative \( f^\prime(x)=9x^2\geq0 \), and only zero at \( x = 0 \), so it is one - to - one), and the inverse of a one - to - one function is one - to - one).

Answer:

\( f^{-1}(x) = \sqrt[3]{\frac{x + 2}{3}} \)

Part (b)