Question
The function \( f \) is defined as \( f(x)=6 x^{2}-7 x \). Find \( f(x-2) \). Write your answer without parentheses, and simplify it as much as pos \( f(x-2)=\square \)
Ask by Sandoval Rose. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f(x-2) = 6x^2 - 31x + 38 \)
Solution
1. Substitute \(x-2\) for \(x\) in the function:
\[
f(x-2)=6(x-2)^2 - 7(x-2)
\]
2. Expand \((x-2)^2\):
\[
(x-2)^2 = x^2 - 4x + 4
\]
So,
\[
f(x-2)=6(x^2 - 4x + 4) - 7(x-2)
\]
3. Distribute the coefficients:
\[
6(x^2 - 4x + 4) = 6x^2 - 24x + 24
\]
\[
-7(x-2) = -7x + 14
\]
4. Combine like terms:
\[
f(x-2)=6x^2 - 24x + 24 - 7x + 14 = 6x^2 - 31x + 38
\]
\( f(x-2)=6x^2-31x+38 \)
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Simplify this solution Bonus Knowledge
To find \( f(x-2) \), we substitute \( x-2 \) into the original function \( f(x) = 6x^2 - 7x \). First, calculate \( f(x-2) \): \[ f(x-2) = 6(x-2)^2 - 7(x-2) \] Now, expand \( (x-2)^2 \): \[ (x-2)^2 = x^2 - 4x + 4 \] Substituting this back into \( f(x-2) \): \[ f(x-2) = 6(x^2 - 4x + 4) - 7(x - 2) \] Now distribute the \( 6 \) and \( -7 \): \[ = 6x^2 - 24x + 24 - 7x + 14 \] Combine like terms: \[ = 6x^2 - 31x + 38 \] Therefore, the simplified expression for \( f(x-2) \) is: \[ f(x-2) = 6x^2 - 31x + 38 \]
