Question
Given the function \( f \), find \( f(-3), f(3), f(-a),-f(a), f(a+h) \) \[ \begin{aligned} f(x) & =2 x-7 \\ f(-3) & =13 \\ f(3) & =-1 \\ f(-a) & =2 a-7 \\ -f(a) & =2 a+7 \\ f(a+h) & =2 a+2 h-7\end{aligned} \]
Ask by Cervantes Vaughan. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\[
\begin{aligned}
f(-3) &= -13, \\
f(3) &= -1, \\
f(-a) &= -2a - 7, \\
-f(a) &= -2a + 7, \\
f(a+h) &= 2a + 2h - 7.
\end{aligned}
\]
Solution
We are given the function
\[
f(x)=2x-7.
\]
1. To find \(f(-3)\), substitute \(x=-3\):
\[
f(-3)=2(-3)-7=-6-7=-13.
\]
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. To find \(-f(a)\), first compute \(f(a)\) and then negate it. For \(x=a\) we have:
\[
f(a)=2a-7.
\]
Thus,
\[
-f(a)=-\left(2a-7\right)=-2a+7.
\]
5. To find \(f(a+h)\), substitute \(x=a+h\):
\[
f(a+h)=2(a+h)-7=2a+2h-7.
\]
The results are:
\[
\begin{aligned}
f(-3)&=-13,\\[1mm]
f(3)&=-1,\\[1mm]
f(-a)&=-2a-7,\\[1mm]
-f(a)&=-2a+7,\\[1mm]
f(a+h)&=2a+2h-7.
\end{aligned}
\]
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Simplify this solution Beyond the Answer
To find the requested values from the function \( f(x) = 2x - 7 \), we can compute them step-by-step: Starting with \( f(-3) \): \[ f(-3) = 2(-3) - 7 = -6 - 7 = -13 \] Next, for \( f(3) \): \[ f(3) = 2(3) - 7 = 6 - 7 = -1 \] Now, for \( f(-a) \): \[ f(-a) = 2(-a) - 7 = -2a - 7 \] For \(-f(a)\): \[ f(a) = 2a - 7 \] Therefore, \[ -f(a) = -(2a - 7) = -2a + 7 \] Lastly, for \( f(a+h) \): \[ f(a+h) = 2(a+h) - 7 = 2a + 2h - 7 \] Putting all the results together: - \( f(-3) = -13 \) - \( f(3) = -1 \) - \( f(-a) = -2a - 7 \) - \( -f(a) = -2a + 7 \) - \( f(a+h) = 2a + 2h - 7 \) Thus, we have the final results for each of the function outputs.
