Question
Evaluate the function at the given value of the independent variable and simplify. 7) \( f(x)=3 x^{2}-3 x-4 ; \quad f(x-1) \)
Ask by Curry French. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f(x-1) = 3x^2 - 9x + 2 \)
Solution
Step 1: Substitute \( x-1 \) in place of \( x \) in the function:
\[
f(x-1)=3(x-1)^2-3(x-1)-4
\]
Step 2: Expand the squared term:
\[
(x-1)^2 = x^2-2x+1
\]
Substitute into the function:
\[
f(x-1)=3(x^2-2x+1)-3(x-1)-4
\]
Step 3: Distribute the constants:
\[
3(x^2-2x+1)=3x^2-6x+3
\]
\[
-3(x-1)=-3x+3
\]
So,
\[
f(x-1)=3x^2-6x+3-3x+3-4
\]
Step 4: Combine like terms:
\[
3x^2-6x-3x=3x^2-9x
\]
\[
3+3-4=2
\]
Hence,
\[
f(x-1)=3x^2-9x+2
\]
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Simplify this solution The Deep Dive
To evaluate the function \( f(x) = 3x^2 - 3x - 4 \) at \( f(x-1) \), substitute \( (x-1) \) into the function: \[ f(x-1) = 3(x-1)^2 - 3(x-1) - 4 \] Now, simplify each term: 1. Calculate \( (x-1)^2 \): \[ (x-1)^2 = x^2 - 2x + 1 \] 2. Substitute that back: \[ f(x-1) = 3(x^2 - 2x + 1) - 3(x - 1) - 4 \] 3. Distribute: \[ = 3x^2 - 6x + 3 - 3x + 3 - 4 \] 4. Combine like terms: \[ = 3x^2 - 9x + 2 \] So, the simplified form of \( f(x-1) \) is \( 3x^2 - 9x + 2 \).
