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

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:** \( f(-x) = x^{2} - 9x + 1 \)