https://r2.erweima.ai/mathbot/image/1726091189086x8ggmxwj.jpg

Let's evaluate each expression based on the graph and tables provided:

### Given:
- \( y = f(x) \) is shown on the graph.
- \( g(x) \) values are provided in the table.
- \( h(x) = 10x - 7 \).

### a) \( f(2) \):
From the graph, when \( x = 2 \), the value of \( f(x) \) is 3.
Thus, \( f(2) = 3 \).

### b) \( g(4) + f(4) \):
- From the table, \( g(4) = -6 \).
- From the graph, when \( x = 4 \), the value of \( f(x) \) is 2.

Therefore,
\[
g(4) + f(4) = -6 + 2 = -4
\]

### c) \( h(2) - g(2) \):
- \( h(x) = 10x - 7 \). For \( x = 2 \),
  \[
  h(2) = 10(2) - 7 = 20 - 7 = 13
  \]
- From the table, \( g(2) = 3 \).

Therefore,
\[
h(2) - g(2) = 13 - 3 = 10
\]

### d) \( f(4.5) \):
From the graph, \( f(4.5) \) lies between \( f(4) = 2 \) and \( f(5) = 0 \). The function appears linear between these points, so we can interpolate:
\[
f(4.5) = \frac{f(5) + f(4)}{2} = \frac{0 + 2}{2} = 1
\]

### Final Answers:
- a) \( f(2) = 3 \)
- b) \( g(4) + f(4) = -4 \)
- c) \( h(2) - g(2) = 10 \)
- d) \( f(4.5) = 1 \)

Would you like further explanations on any of these steps or details? Here are 5 related questions to expand on:

1. How do you interpolate between two points on a graph?
2. What is the general formula for a linear function?
3. How can you determine \( f(x) \) for values not listed on a graph?
4. What is the process of evaluating piecewise functions like \( f(x) \)?
5. How do you solve a system involving multiple functions like \( f(x) \), \( g(x) \), and \( h(x) \)?

**Tip:** When using a graph to evaluate function values, double-check the corresponding \( x \)- and \( y \)-coordinates carefully for accuracy.

Evaluate the following expressions based on the graph and tables: a) f(2), b) g(4) + f(4), c) h(2) - g(2), d) f(4.5).

Learn how to evaluate expressions for f(x), g(x), and h(x) using a combination of graph interpretation and table data. This problem covers key algebraic concepts such as linear interpolation and function evaluation.

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