Sure, let's evaluate the function [tex]\( f(x) = x^2 - 4x + 2 \)[/tex] at the specified values step by step.
### (a) Evaluate [tex]\( f(-1) \)[/tex]:
We start by substituting [tex]\( -1 \)[/tex] into the function for [tex]\( x \)[/tex]:
[tex]\[
f(-1) = (-1)^2 - 4(-1) + 2
\][/tex]
Next, we calculate each term:
[tex]\[
(-1)^2 = 1
\][/tex]
[tex]\[
-4(-1) = 4
\][/tex]
Adding these values together with the constant term:
[tex]\[
f(-1) = 1 + 4 + 2 = 7
\][/tex]
So,
[tex]\[
f(-1) = 7
\][/tex]
### (b) Evaluate [tex]\( f\left(\frac{1}{2}\right) \)[/tex]:
We substitute [tex]\( \frac{1}{2} \)[/tex] into the function for [tex]\( x \)[/tex]:
[tex]\[
f\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^2 - 4\left(\frac{1}{2}\right) + 2
\][/tex]
Next, we calculate each term:
[tex]\[
\left(\frac{1}{2}\right)^2 = \frac{1}{4}
\][/tex]
[tex]\[
-4\left(\frac{1}{2}\right) = -2
\][/tex]
Adding these values together with the constant term:
[tex]\[
f\left(\frac{1}{2}\right) = \frac{1}{4} - 2 + 2 = \frac{1}{4}
\][/tex]
So,
[tex]\[
f\left(\frac{1}{2}\right) = 0.25
\][/tex]
### (c) Evaluate [tex]\( f(c+2) \)[/tex]:
We now substitute [tex]\( c+2 \)[/tex] into the function for [tex]\( x \)[/tex]:
[tex]\[
f(c+2) = (c+2)^2 - 4(c+2) + 2
\][/tex]
First, we expand [tex]\((c+2)^2\)[/tex]:
[tex]\[
(c+2)^2 = c^2 + 4c + 4
\][/tex]
Next, we distribute [tex]\(-4\)[/tex] across [tex]\((c+2)\)[/tex]:
[tex]\[
-4(c+2) = -4c - 8
\][/tex]
Combining these results with the constant term [tex]\( +2 \)[/tex]:
[tex]\[
f(c+2) = c^2 + 4c + 4 - 4c - 8 + 2
\][/tex]
Simplifying by combining like terms:
[tex]\[
f(c+2) = c^2 + 4 - 8 + 2
\][/tex]
[tex]\[
f(c+2) = c^2 - 2
\][/tex]
So, the simplified expression is:
[tex]\[
f(c+2) = c^2 - 2
\][/tex]
Thus, the evaluations are:
(a) [tex]\( f(-1) = 7 \)[/tex]
(b) [tex]\( f\left(\frac{1}{2}\right) = 0.25 \)[/tex]
(c) [tex]\( f(c+2) = c^2 - 2 \)[/tex]
