Evaluate the function $$ f(x)=x^{2}+9 x+1 $$ at the given values of the independent variable and simplify. $$ \begin{array}{lll} ext { a. } f(-8) & ext { b. } f(x+5) & ext { c. } f(-x)\end{array} $$

Question

Evaluate the function $$ f(x)=x^{2}+9 x+1 $$ at the given values of the independent variable and simplify. $$ \begin{array}{lll}	ext { a. } f(-8) & 	ext { b. } f(x+5) & 	ext { c. } f(-x)\end{array} $$

UpStudy AI · Accepted Answer

Let's evaluate the function $$ f(x) = x^{2} + 9x + 1 $$ for the given inputs.

### a. $$ f(-8) $$

To find $$ f(-8) $$, substitute $$ x = -8 $$ into the function:

$$
f(-8) = (-8)^2 + 9(-8) + 1
$$

Calculate each term:

$$
(-8)^2 = 64
$$
$$
9(-8) = -72
$$
$$
1 = 1
$$

Now, add them together:

$$
f(-8) = 64 - 72 + 1 = (-8) + 1 = -7
$$

**Answer:** $$ f(-8) = -7 $$

---

### b. $$ f(x+5) $$

To find $$ f(x+5) $$, substitute $$ x+5 $$ into the function wherever there is an $$ x $$:

$$
f(x+5) = (x + 5)^2 + 9(x + 5) + 1
$$

Expand and simplify each term:

1. Expand $$ (x + 5)^2 $$:

$$
(x + 5)^2 = x^2 + 10x + 25
$$

2. Expand $$ 9(x + 5) $$:

$$
9(x + 5) = 9x + 45
$$

3. Add all the terms together:

$$
f(x+5) = x^2 + 10x + 25 + 9x + 45 + 1
$$

Combine like terms:

$$
x^2 + (10x + 9x) + (25 + 45 + 1) = x^2 + 19x + 71
$$

**Answer:** $$ f(x+5) = x^{2} + 19x + 71 $$

---

### c. $$ f(-x) $$

To find $$ f(-x) $$, substitute $$ -x $$ into the function wherever there is an $$ x $$:

$$
f(-x) = (-x)^2 + 9(-x) + 1
$$

Calculate each term:

$$
(-x)^2 = x^2
$$
$$
9(-x) = -9x
$$
$$
1 = 1
$$

Now, combine them:

$$
f(-x) = x^2 - 9x + 1
$$

**Answer:** $$ f(-x) = x^{2} - 9x + 1 $$
a. $$ f(-8) = -7 $$ b. $$ f(x+5) = x^{2} + 19x + 71 $$ c. $$ f(-x) = x^{2} - 9x + 1 $$

Evaluate the function \( f(x)=x^{2}+9 x+1 \) at the given values of the independent variable and simplify. \( \begin{array}{lll}\text { a. } f(-8) & \text { b. } f(x+5) & \text { c. } f(-x)\end{array} \)

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