https://file.aiquickdraw.com/mathbot/file/1727061336948h56f6gv4.jpg

Let's work through both parts of this problem:

### a) \( f(x) = x^2 - 4 \)

1. **Finding the inverse of \( f(x) \):**
   - To find the inverse, replace \( f(x) \) with \( y \): 
     \[
     y = x^2 - 4
     \]
   - Swap \( x \) and \( y \) and solve for \( y \):
     \[
     x = y^2 - 4
     \]
     \[
     y^2 = x + 4
     \]
     \[
     y = \pm \sqrt{x + 4}
     \]
   Thus, the inverse relation of \( f(x) \) is \( f^{-1}(x) = \pm \sqrt{x + 4} \).

   - **Graphing**: 
     - The original function \( f(x) = x^2 - 4 \) is a parabola opening upwards, with a vertex at \( (0, -4) \).
     - The inverse relation \( f^{-1}(x) = \pm \sqrt{x + 4} \) is not a function, since it gives two values (positive and negative square roots) for each \( x \), except when \( x = -4 \).

   - **Is the inverse a function?** No, the inverse is not a function because it does not pass the vertical line test (it provides two outputs for one input).

---

### b) \( g(x) = 2^x \)

1. **Finding the inverse of \( g(x) \):**
   - Replace \( g(x) \) with \( y \): 
     \[
     y = 2^x
     \]
   - Swap \( x \) and \( y \), and solve for \( y \):
     \[
     x = 2^y
     \]
     \[
     y = \log_2(x)
     \]
   Thus, the inverse function is \( g^{-1}(x) = \log_2(x) \).

   - **Graphing**:
     - The original function \( g(x) = 2^x \) is an exponential curve that increases rapidly for positive \( x \) and approaches zero as \( x \) becomes more negative.
     - The inverse function \( g^{-1}(x) = \log_2(x) \) is a logarithmic curve that grows slowly for larger values of \( x \) and is undefined for \( x \leq 0 \).

   - **Is the inverse a function?** Yes, the inverse of \( g(x) \) is a function because it passes the vertical line test.

---

Do you need help with a graphical representation, or do you have any specific questions about the solutions?

Here are 5 questions to further your understanding:
1. What does the vertical line test indicate about a relation being a function?
2. How do we algebraically find the inverse of a quadratic function?
3. Why is the inverse of an exponential function always a logarithmic function?
4. What restrictions exist on the domain of \( f(x) = x^2 - 4 \) and its inverse?
5. Can every function have an inverse that is also a function? Why or why not?

**Tip:** Always check if the inverse relation is a function by applying the vertical line test to its graph.

Graph each function and its inverse relation on the same set of axes. Determine whether the inverse relation is a function. a) f(x) = x^2 - 4 b) g(x) = 2^x

Explore how to graph the functions f(x) = x^2 - 4 and g(x) = 2^x along with their inverses. Learn to determine if the inverse relation is a function using the vertical line test. This lesson covers key concepts in graphing and inverse functions, suitable for students in Grades 10-12.

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