https://file.aiquickdraw.com/mathbot/file/1727117095774yhmb0sjk.jpg

Let's begin by solving the functions as requested in the image. We will handle each problem step by step.

### 1. \( f(x) = 2(3x - 6) + 7 \)
- **To find \( f(7) \)**:
  \[
  f(7) = 2(3(7) - 6) + 7 = 2(21 - 6) + 7 = 2(15) + 7 = 30 + 7 = 37
  \]
- **To find \( f(-3) \)**:
  \[
  f(-3) = 2(3(-3) - 6) + 7 = 2(-9 - 6) + 7 = 2(-15) + 7 = -30 + 7 = -23
  \]

### 2. \( g(x) = \frac{7(3x + 5)}{x} + 5 \)
- **To find \( g(2) \)**:
  \[
  g(2) = \frac{7(3(2) + 5)}{2} + 5 = \frac{7(6 + 5)}{2} + 5 = \frac{7(11)}{2} + 5 = \frac{77}{2} + 5 = 38.5 + 5 = 43.5
  \]
- **To find \( g(0) \)**:
  Substituting \( x = 0 \) causes a division by zero, so \( g(0) \) is undefined.
- **To find \( g(-5) \)**:
  \[
  g(-5) = \frac{7(3(-5) + 5)}{-5} + 5 = \frac{7(-15 + 5)}{-5} + 5 = \frac{7(-10)}{-5} + 5 = \frac{-70}{-5} + 5 = 14 + 5 = 19
  \]

### 3. \( f(x) = \frac{2x}{5} + \frac{8(4x - 7)}{2x} \)
- **To find \( f(-2) \)**:
  \[
  f(-2) = \frac{2(-2)}{5} + \frac{8(4(-2) - 7)}{2(-2)} = \frac{-4}{5} + \frac{8(-8 - 7)}{-4} = \frac{-4}{5} + \frac{8(-15)}{-4} = \frac{-4}{5} + 30 = 30 - \frac{4}{5} = 29.2
  \]
- **To find \( f(6) \)**:
  \[
  f(6) = \frac{2(6)}{5} + \frac{8(4(6) - 7)}{2(6)} = \frac{12}{5} + \frac{8(24 - 7)}{12} = \frac{12}{5} + \frac{8(17)}{12} = \frac{12}{5} + \frac{136}{12} = 2.4 + 11.33 = 13.73
  \]

### 4. \( c(x) = 5(1.23)^{3x} \)
- **To find \( c(-2) \)**:
  \[
  c(-2) = 5(1.23)^{3(-2)} = 5(1.23)^{-6} \approx 5(0.6508) = 3.254
  \]
- **To find \( c(1) \)**:
  \[
  c(1) = 5(1.23)^{3(1)} = 5(1.23)^3 \approx 5(1.860867) = 9.304
  \]

### 5. \( f(x) = 3x^{-2} \)
- **To find \( f(-1) \)**:
  \[
  f(-1) = 3(-1)^{-2} = 3(1) = 3
  \]
- **To find \( f(0) \)**:
  Since \( x = 0 \) would cause division by zero, \( f(0) \) is undefined.
- **To find \( f(5) \)**:
  \[
  f(5) = 3(5)^{-2} = 3 \times \frac{1}{25} = \frac{3}{25} = 0.12
  \]

### Graphing (Questions 6 and 7)
For questions 6 and 7, you need to graph the functions from problems 1 and 4:
- **Question 6**: Graph \( f(x) = 2(3x - 6) + 7 \)
- **Question 7**: Graph \( c(x) = 5(1.23)^{3x} \)

Both graphs can be plotted either manually or by using Excel by inputting values of \( x \) and calculating corresponding values of \( f(x) \) and \( c(x) \).

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Would you like detailed instructions on how to graph these in Excel or any further explanation of the computations?

### Further Related Questions:
1. How do you simplify and solve for \( f(x) \) if a new function is given with fractional exponents?
2. What happens when division by zero occurs in a function, and how do we address undefined points in a graph?
3. Can we express \( f(x) = 5(1.23)^{3x} \) using logarithmic transformations to linearize it for graphing?
4. What techniques can be used to solve equations involving negative exponents like \( 3x^{-2} \)?
5. How can you analyze the behavior of a function as \( x \to \infty \) or \( x \to -\infty \)?

### Tip:
When using Excel to graph functions, you can utilize the "Scatter with Smooth Lines" feature for better visualization of continuous functions.

Evaluate the functions in 1 – 5 as indicated by hand and by using Excel.

Solve algebraic functions involving linear, rational, and exponential terms. Find specific function values and handle division by zero. Graph functions by hand and in Excel for visualization.

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