C. Find ( fog) (x), (gof) (x), and (fog) (3). Show your solution.
1.  f(x)=5x-3, 	g(x)=x2 -3

To solve the problem, we need to find the compositions \((f \circ g)(x)\), \((g \circ f)(x)\), and \((f \circ g)(3)\) given the functions \(f(x) = 5x - 3\) and \(g(x) = x^2 - 3\).

### 1. Finding \((f \circ g)(x)\):
The composition \((f \circ g)(x)\) means \(f(g(x))\), which means we substitute \(g(x)\) into \(f(x)\).

Given:
\[ g(x) = x^2 - 3 \]
\[ f(x) = 5x - 3 \]

Substitute \(g(x)\) into \(f(x)\):
\[ (f \circ g)(x) = f(g(x)) = f(x^2 - 3) \]

Now, apply the function \(f\) to \(x^2 - 3\):
\[ f(x^2 - 3) = 5(x^2 - 3) - 3 \]

Distribute the 5:
\[ f(x^2 - 3) = 5x^2 - 15 - 3 \]

Combine like terms:
\[ (f \circ g)(x) = 5x^2 - 18 \]

### 2. Finding \((g \circ f)(x)\):
The composition \((g \circ f)(x)\) means \(g(f(x))\), which means we substitute \(f(x)\) into \(g(x)\).

Given:
\[ f(x) = 5x - 3 \]
\[ g(x) = x^2 - 3 \]

Substitute \(f(x)\) into \(g(x)\):
\[ (g \circ f)(x) = g(f(x)) = g(5x - 3) \]

Now, apply the function \(g\) to \(5x - 3\):
\[ g(5x - 3) = (5x - 3)^2 - 3 \]

Expand \((5x - 3)^2\):
\[ (5x - 3)^2 = 25x^2 - 30x + 9 \]

So,
\[ g(5x - 3) = 25x^2 - 30x + 9 - 3 \]

Combine like terms:
\[ (g \circ f)(x) = 25x^2 - 30x + 6 \]

### 3. Finding \((f \circ g)(3)\):
To find \((f \circ g)(3)\), we substitute \(x = 3\) into \((f \circ g)(x)\).

We already found:
\[ (f \circ g)(x) = 5x^2 - 18 \]

Now substitute \(x = 3\):
\[ (f \circ g)(3) = 5(3^2) - 18 \]

Calculate \(3^2\):
\[ (f \circ g)(3) = 5(9) - 18 \]

Multiply:
\[ (f \circ g)(3) = 45 - 18 \]

Subtract:
\[ (f \circ g)(3) = 27 \]

### Final Answers:
1. \((f \circ g)(x) = 5x^2 - 18\)
2. \((g \circ f)(x) = 25x^2 - 30x + 6\)
3. \((f \circ g)(3) = 27\)

Would you like more details or have any questions? Here are some related questions for further practice:

1. What is the domain of \((f \circ g)(x)\) and \((g \circ f)(x)\)?
2. How would you find \((f \circ f)(x)\) and \((g \circ g)(x)\)?
3. Calculate \((f \circ g)(2)\) using the same functions.
4. What is the range of \((f \circ g)(x)\)?
5. How does the composition \((f \circ g)(x)\) change if \(g(x)\) were different?
6. Can you find the inverse of \((f \circ g)(x)\)?
7. How do you determine whether \((f \circ g)(x)\) is a one-to-one function?
8. What are the zeros of \((f \circ g)(x)\)?

**Tip:** When finding compositions of functions, always substitute the inner function first before applying the outer function.

Explore how to solve (f ∘ g)(x), (g ∘ f)(x), and (f ∘ g)(3) using given functions f(x) = 5x - 3 and g(x) = x^2 - 3. Understand the step-by-step process with detailed examples and calculations. Suitable for high school students learning composition of functions and quadratic equations.

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