Given the function f left parenthesis x right parenthesis equals x cubed minus 3
​,
​(a) Find f Superscript negative 1 Baseline left parenthesis x right parenthesis
.
​(b) Graph f and f Superscript negative 1
 
in the same rectangular coordinate system.
​(c) Use interval notation to give the domain and the range of f and f Superscript negative 1
.
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Part 1
​(a) Find f Superscript negative 1 Baseline left parenthesis x right parenthesis
.
f Superscript negative 1 Baseline left parenthesis x right parenthesis equals nothing


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.

Given the function f(x) = x^3 - 3, (a) Find f⁻¹(x). (b) Graph f and f⁻¹ in the same rectangular coordinate system. (c) Use interval notation to give the domain and the range of f and f⁻¹.

Solve a function problem by finding the inverse of f(x) = x^3 - 3 and graphing both f and f⁻¹ in the same coordinate system. Learn how to use interval notation to describe the domain and range of these functions. Suitable for Grades 10-12.

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