Question

Find all values of x satisfying the given conditions.
f left parenthesis x right parenthesis equals 3 x minus 6 comma g left parenthesis x right parenthesis equals x squared minus 6 x plus 20 comma and left parenthesis f circle g right parenthesis left parenthesis x right parenthesis equals 30
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Part 1
The values of x satisfying the given conditions are
  
enter your response here.
​(Use a comma to separate answers as​ needed.)

Answer 1

To find the values of \( x \) that satisfy the condition \( (f \circ g)(x) = 30 \), we will start with the functions defined in the problem:

- \( f(x) = 3x - 6 \)
- \( g(x) = x^2 - 6x + 20 \)

The composition of functions \( (f \circ g)(x) \) means we need to evaluate \( f(g(x)) \). We can calculate this as follows:

1. **Evaluate \( g(x) \)**:
\[
g(x) = x^2 - 6x + 20
\]

2. **Substitute \( g(x) \) into \( f \)**:
\[
f(g(x)) = f(x^2 - 6x + 20) = 3(x^2 - 6x + 20) - 6
\]

3. **Expand \( f(g(x)) \)**:
\[
f(g(x)) = 3x^2 - 18x + 60 - 6 = 3x^2 - 18x + 54
\]

4. **Set \( f(g(x)) \) equal to 30**:
\[
3x^2 - 18x + 54 = 30
\]

5. **Subtract 30 from both sides**:
\[
3x^2 - 18x + 54 - 30 = 0 \implies 3x^2 - 18x + 24 = 0
\]

6. **Divide the entire equation by 3**:
\[
x^2 - 6x + 8 = 0
\]

7. **Factor the quadratic equation**:
\[
(x - 2)(x - 4) = 0
\]

8. **Solve for \( x \)**:
\[
x = 2 \quad \text{or} \quad x = 4
\]

The values of \( x \) satisfying the given conditions are:

\[
\boxed{2, 4}
\]