Let's analyze the question in the image you provided.

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: $f^{-1}(x) = \sqrt{\frac{-x}{2}}$

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.