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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).

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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.

For Exercises 51-56, suppose f is a function and a function g is defined by the given expression. (a) Write g as the composition of f and one or two linear functions. (b) Describe how the graph of g is obtained from the graph of f. 51. g(x) = 3f(x) - 2 54. g(x) = f(-2/3x)

Explore how to compose functions and transform graphs with the examples g(x) = 3f(x) - 2 and g(x) = f(-2/3x). This problem covers key concepts such as vertical stretching, shifting, horizontal stretching, and reflection. Suitable for Grades 10-12.

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