Question
Given the function \( f \), find \( f(-3), f(3), f(-a),-f(a), f(a+h) \) \[ \begin{array}{l}f(x)=2 x-7 \\ f(-3)= \\ f(3)= \\ f(-a)= \\ -f(a)= \\ f(a+h)= \\ \end{array} \]
Ask by Reeves Chang. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f(-3) = -13 \), \( f(3) = -1 \), \( f(-a) = -2a - 7 \), \( -f(a) = -2a + 7 \), \( f(a+h) = 2a + 2h - 7 \).
Solution
1. To find \( f(-3) \), substitute \( x=-3 \) into \( f(x)=2x-7 \):
\[
f(-3)=2(-3)-7=-6-7=-13.
\]
2. To find \( f(3) \), substitute \( x=3 \):
\[
f(3)=2(3)-7=6-7=-1.
\]
3. To find \( f(-a) \), substitute \( x=-a \):
\[
f(-a)=2(-a)-7=-2a-7.
\]
4. Given \( f(a)=2a-7 \), the expression for \( -f(a) \) is:
\[
-f(a)=-\left(2a-7\right)=-2a+7.
\]
5. To find \( f(a+h) \), substitute \( x=a+h \) into the function:
\[
f(a+h)=2(a+h)-7=2a+2h-7.
\]
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Simplify this solution Beyond the Answer
Let's evaluate the function \( f(x) = 2x - 7 \) for the different inputs. First, we'll find \( f(-3) \): \[ f(-3) = 2(-3) - 7 = -6 - 7 = -13 \] Next, we compute \( f(3) \): \[ f(3) = 2(3) - 7 = 6 - 7 = -1 \] Now, we calculate \( f(-a) \): \[ f(-a) = 2(-a) - 7 = -2a - 7 \] For \( -f(a) \): \[ f(a) = 2a - 7, \quad \text{so} \quad -f(a) = -(2a - 7) = -2a + 7 \] Lastly, we find \( f(a+h) \): \[ f(a+h) = 2(a+h) - 7 = 2a + 2h - 7 \] In summary: \[ f(-3) = -13, \quad f(3) = -1, \quad f(-a) = -2a - 7, \quad -f(a) = -2a + 7, \quad f(a+h) = 2a + 2h - 7 \]
