Sure! Let's evaluate the function [tex]\( f(x) = 3x^2 - 5x \)[/tex] at the specified values step-by-step.
### 1. Evaluate [tex]\( f(-1) \)[/tex]
To find [tex]\( f(-1) \)[/tex], substitute [tex]\( x = -1 \)[/tex] into the function:
[tex]\[
f(-1) = 3(-1)^2 - 5(-1)
\][/tex]
Now calculate each term individually:
[tex]\[
3(-1)^2 = 3 \cdot 1 = 3
\][/tex]
[tex]\[
-5(-1) = 5
\][/tex]
Combine these results:
[tex]\[
f(-1) = 3 + 5 = 8
\][/tex]
### 2. Evaluate [tex]\( f(1) \)[/tex]
To find [tex]\( f(1) \)[/tex], substitute [tex]\( x = 1 \)[/tex] into the function:
[tex]\[
f(1) = 3(1)^2 - 5(1)
\][/tex]
Now calculate each term individually:
[tex]\[
3(1)^2 = 3 \cdot 1 = 3
\][/tex]
[tex]\[
-5(1) = -5
\][/tex]
Combine these results:
[tex]\[
f(1) = 3 - 5 = -2
\][/tex]
### 3. Evaluate [tex]\( f(a) \)[/tex]
To find [tex]\( f(a) \)[/tex], substitute [tex]\( x = a \)[/tex] into the function:
[tex]\[
f(a) = 3a^2 - 5a
\][/tex]
### 4. Evaluate [tex]\( -f(a) \)[/tex]
To find [tex]\( -f(a) \)[/tex], negate the expression for [tex]\( f(a) \)[/tex]:
[tex]\[
-f(a) = -(3a^2 - 5a)
\][/tex]
Distribute the negative sign:
[tex]\[
-f(a) = -3a^2 + 5a
\][/tex]
### 5. Evaluate [tex]\( f(a + h) \)[/tex]
To find [tex]\( f(a + h) \)[/tex], substitute [tex]\( x = a + h \)[/tex] into the function:
[tex]\[
f(a + h) = 3(a + h)^2 - 5(a + h)
\][/tex]
Expand the squared term:
[tex]\[
(a + h)^2 = a^2 + 2ah + h^2
\][/tex]
So,
[tex]\[
3(a + h)^2 = 3(a^2 + 2ah + h^2) = 3a^2 + 6ah + 3h^2
\][/tex]
Distribute and combine all terms:
[tex]\[
-5(a + h) = -5a - 5h
\][/tex]
Combine these results together:
[tex]\[
f(a + h) = 3a^2 + 6ah + 3h^2 - 5a - 5h
\][/tex]
Thus, the evaluation of the function [tex]\( f(x) = 3x^2 - 5x \)[/tex] at the specified values provides the following results:
1. [tex]\( f(-1) = 8 \)[/tex]
2. [tex]\( f(1) = -2 \)[/tex]
3. [tex]\( f(a) = 3a^2 - 5a \)[/tex]
4. [tex]\( -f(a) = -3a^2 + 5a \)[/tex]
5. [tex]\( f(a + h) = 3a^2 + 6ah + 3h^2 - 5a - 5h \)[/tex]
