Let's break down the problems shown in the image, which ask us to express the function $g(x)$ as a composition of linear functions and describe how the graph of $g(x)$ relates to that of $f(x)$.

Problem 51: $g(x) = 3f(x) - 2$

(a) Write $g(x)$ as the composition of $f$ and one or two linear functions.

We can rewrite $g(x)$ as:

$g(x) = 3f(x) - 2$

This can be broken down into two steps, involving linear transformations:

1. Multiply the output of $f(x)$ by 3.
2. Subtract 2 from the result.

Thus, we can express this as a composition of functions:

$g(x) = h(f(x)), \quad \text{where} \ h(x) = 3x - 2$

(b) Describe how the graph of $g(x)$ is obtained from the graph of $f(x)$.

• Multiplying by 3 stretches the graph of $f(x)$ vertically by a factor of 3.
• Subtracting 2 shifts the graph down by 2 units.

Problem 54: $g(x) = f\left(-\frac{2}{3}x\right)$

(a) Write $g(x)$ as the composition of $f$ and one or two linear functions.

In this case:

$g(x) = f\left(-\frac{2}{3}x\right)$

We can interpret this as a linear transformation on the input before applying $f$. The transformation can be broken down as:

1. Multiply $x$ by $-\frac{2}{3}$.

Thus, we have:

$g(x) = f(k(x)), \quad \text{where} \ k(x) = -\frac{2}{3}x$

(b) Describe how the graph of $g(x)$ is obtained from the graph of $f(x)$.

• Multiplying the input by $-\frac{2}{3}$ causes a horizontal stretch by a factor of $\frac{3}{2}$ (since $\frac{1}{-\frac{2}{3}} = \frac{3}{2}$) and a reflection across the $y$-axis (due to the negative sign).

Would you like to dive into more details on these transformations or the process?

Here are 5 related questions to explore further:

1. How does multiplying a function by a constant affect its graph?
2. What happens when you reflect a function across the $y$-axis?
3. How does vertical shifting compare to horizontal shifting?
4. Can any non-linear transformations be composed similarly to linear ones?
5. What are the key differences between stretching a graph horizontally and vertically?

Tip: When transforming functions, always consider how changes in input affect the graph's horizontal features, while changes in the output modify vertical aspects.