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The functions given are:
- \( f(x) = bx^2 \)
- \( g(x) = a^x \)

From the graph, we need to determine values for \( a \) and \( b \), as well as solve additional parts.

### 1.1 Determine the values of \( a \) and \( b \).

#### \( f(x) = bx^2 \)
From the graph, for the parabola \( f(x) \), it passes through the point \( (-2, -8) \). Substituting this into the equation of \( f(x) \):
\[
f(-2) = b(-2)^2 = -8
\]
\[
4b = -8 \quad \Rightarrow \quad b = -2
\]
Thus, \( f(x) = -2x^2 \).

#### \( g(x) = a^x \)
For the exponential function \( g(x) = a^x \), it passes through the point \( (2, 9) \). Substituting into the equation of \( g(x) \):
\[
g(2) = a^2 = 9
\]
\[
a^2 = 9 \quad \Rightarrow \quad a = 3
\]
Thus, \( g(x) = 3^x \).

### 1.2 Write a possible restriction for \( f(x) \) so that the inverse of \( f(x) \) will be a function.
Since \( f(x) = -2x^2 \) is a parabola, to ensure its inverse is a function, we need to restrict \( f(x) \) to either the left or right half of the parabola. For example, we can restrict \( x \geq 0 \) or \( x \leq 0 \). A possible restriction would be:
\[
x \geq 0
\]
This ensures that \( f(x) \) is one-to-one and its inverse will be a function.

### 1.3 Determine the equation of the inverse function of \( f(x) \).

To find the inverse of \( f(x) = -2x^2 \) (with the restriction \( x \geq 0 \)):
1. Start with \( y = -2x^2 \)
2. Solve for \( x \) in terms of \( y \):
\[
y = -2x^2 \quad \Rightarrow \quad x^2 = \frac{-y}{2} \quad \Rightarrow \quad x = \sqrt{\frac{-y}{2}}
\]
Since \( x \geq 0 \), the inverse function is:
\[
f^{-1}(x) = \sqrt{\frac{-x}{2}}
\]

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Let me know if you'd like further details or have any specific questions! Here are five related questions you can explore:
1. What if we restrict \( f(x) \) to \( x \leq 0 \)? What would the inverse function be?
2. How does the domain restriction affect the range of the inverse function?
3. Can you graph the inverse function of \( f(x) \)?
4. What are the properties of inverse functions for quadratic equations in general?
5. How would changing the value of \( b \) in \( f(x) \) affect its graph and inverse?

**Tip**: Always remember that for a function to have an inverse, it must be one-to-one, which can often require domain restrictions for functions like quadratics.

The functions y = f(x) = bx^2 and y = g(x) = a^x are given. Determine the values of a and b, write down a possible restriction for f(x) so that its inverse will be a function, and determine the equation of the inverse function of f.

Solve a quadratic and exponential function problem by determining the values of a and b, applying restrictions for f(x) to find its inverse, and calculating the inverse function equation. Includes domain restriction analysis and inverse function calculations. Suitable for students in Grades 10-12.

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