Question

Evaluate the function f left parenthesis x right parenthesis equals x squared plus 7 x plus 8 at the given values of the independent variable and simplify.
a.  f left parenthesis 3 right parenthesis      b.  f left parenthesis x plus 1 right parenthesis      c.  f left parenthesis negative x right parenthesis
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Part 1
a.  f left parenthesis 3 right parenthesisequals
  
enter your response here ​(Simplify your​ answer.)
Part 2
b.  f left parenthesis x plus 1 right parenthesisequals
  
enter your response here ​(Simplify your​ answer.)
Part 3
c.  f left parenthesis negative x right parenthesisequals
  
enter your response here ​(Simplify your​ answer.)

Answer 1

Let's evaluate the function \( f(x) = x^2 + 7x + 8 \) for the given values.

### Part 1: Evaluating \( f(3) \)
To find \( f(3) \):

\[
f(3) = 3^2 + 7(3) + 8
\]

Calculating each term:

\[
3^2 = 9
\]

\[
7(3) = 21
\]

Now substitute back into the equation:

\[
f(3) = 9 + 21 + 8 = 38
\]

**Answer for Part 1:** \( f(3) = 38 \)

### Part 2: Evaluating \( f(x + 1) \)
To find \( f(x + 1) \), we substitute \( x + 1 \) into the function:

\[
f(x + 1) = (x + 1)^2 + 7(x + 1) + 8
\]

Calculating each term:

\[
(x + 1)^2 = x^2 + 2x + 1
\]

\[
7(x + 1) = 7x + 7
\]

Now combine:

\[
f(x + 1) = (x^2 + 2x + 1) + (7x + 7) + 8
\]

Combining all terms:

\[
f(x + 1) = x^2 + 2x + 1 + 7x + 7 + 8
\]

\[
f(x + 1) = x^2 + (2x + 7x) + (1 + 7 + 8)
\]

\[
f(x + 1) = x^2 + 9x + 16
\]

**Answer for Part 2:** \( f(x + 1) = x^2 + 9x + 16 \)

### Part 3: Evaluating \( f(-x) \)
To find \( f(-x) \), we substitute \( -x \) into the function:

\[
f(-x) = (-x)^2 + 7(-x) + 8
\]

Calculating each term:

\[
(-x)^2 = x^2
\]

\[
7(-x) = -7x
\]

Now combine:

\[
f(-x) = x^2 - 7x + 8
\]

**Answer for Part 3:** \( f(-x) = x^2 - 7x + 8 \)

### Summary of Answers:
- Part 1: \( f(3) = 38 \)
- Part 2: \( f(x + 1) = x^2 + 9x + 16 \)
- Part 3: \( f(-x) = x^2 - 7x + 8 \)