Let's tackle the problem step-by-step.

(a) Finding the inverse of the function $f(x) = x^3 - 3$:

To find the inverse $f^{-1}(x)$, we follow these steps:

1. Start with $y = f(x) = x^3 - 3$.
2. Swap $x$ and $y$ to find the inverse: $x = y^3 - 3$
3. Solve for $y$: $x + 3 = y^3$ $y = \sqrt[3]{x + 3}$ Thus, the inverse function is: $f^{-1}(x) = \sqrt[3]{x + 3}$

(b) Graphing $f(x)$ and $f^{-1}(x)$:

The function $f(x) = x^3 - 3$ is a cubic function, which has the characteristic shape of a cubic curve. The inverse function $f^{-1}(x) = \sqrt[3]{x + 3}$ is also a cubic-root function but reflects the inverse relationship.

I can describe the graph:

• $f(x)$ is increasing for all $x$, with a point of inflection at $x = 0$, where $f(0) = -3$.
• $f^{-1}(x)$ is the reflection of $f(x)$ across the line $y = x$.

The graphs of $f(x)$ and $f^{-1}(x)$ should intersect on this line $y = x$. Both functions pass through the point $(-3, 0)$, and their behavior reflects each other symmetrically.

(c) Domain and Range of $f(x)$ and $f^{-1}(x)$:

• For $f(x) = x^3 - 3$:

  • Domain: $(-\infty, \infty)$, because the cubic function is defined for all real numbers.
  • Range: $(-\infty, \infty)$, because cubic functions take all real values.

• For $f^{-1}(x) = \sqrt[3]{x + 3}$:

  • Domain: $(-\infty, \infty)$, since the cube root function is also defined for all real numbers.
  • Range: $(-\infty, \infty)$, because the cube root function also takes all real values.

Would you like a graph for further illustration? Let me know if you want additional details on any part!

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Related questions:

1. How do you find the inverse of more complex functions, such as quadratics?
2. How does symmetry between a function and its inverse manifest graphically?
3. What is the general method for finding the inverse of a function?
4. How do different transformations affect the domain and range of a function?
5. Can every function have an inverse?

Tip:

When finding the inverse of a function, always check that the function is one-to-one by using the horizontal line test. This ensures the inverse is a valid function.