Question

for the function ( f(x) = 4x^2 - 4x ), evaluate and fully simplify each of the following. ( f(x + h) = ) ( \frac{f(x + h) - f(x)}{h} = ) question help: video

Explanation:

Part 1: Find \( f(x + h) \)

Step 1: Substitute \( x + h \) into \( f(x) \)

We have \( f(x)=4x^{2}-4x \). To find \( f(x + h) \), we replace every \( x \) in the function with \( x + h \). So we get:
\( f(x + h)=4(x + h)^{2}-4(x + h) \)

Step 2: Expand \( (x + h)^{2} \)

Using the formula \( (a + b)^{2}=a^{2}+2ab + b^{2} \), where \( a = x \) and \( b = h \), we expand \( (x + h)^{2} \) to \( x^{2}+2xh+h^{2} \). Then:
\( f(x + h)=4(x^{2}+2xh + h^{2})-4(x + h) \)

Step 3: Distribute the 4 and -4

Distribute the 4 into \( (x^{2}+2xh + h^{2}) \) and the -4 into \( (x + h) \):
\( f(x + h)=4x^{2}+8xh + 4h^{2}-4x-4h \)

Part 2: Find \( \frac{f(x + h)-f(x)}{h} \)

Step 1: Substitute \( f(x + h) \) and \( f(x) \)

We know \( f(x + h)=4x^{2}+8xh + 4h^{2}-4x-4h \) and \( f(x)=4x^{2}-4x \). Substitute these into the expression:
\( \frac{(4x^{2}+8xh + 4h^{2}-4x-4h)-(4x^{2}-4x)}{h} \)

Step 2: Simplify the numerator

Remove the parentheses in the numerator:
\( \frac{4x^{2}+8xh + 4h^{2}-4x-4h - 4x^{2}+4x}{h} \)
Now, combine like terms. The \( 4x^{2} \) and \( -4x^{2} \) cancel out, and the \( -4x \) and \( +4x \) cancel out:
\( \frac{8xh + 4h^{2}-4h}{h} \)

Step 3: Factor out \( h \) from the numerator

Factor \( h \) from each term in the numerator:
\( \frac{h(8x + 4h-4)}{h} \)

Step 4: Cancel out \( h \) (assuming \( h

eq0 \))
Cancel the common factor of \( h \) in the numerator and the denominator:
\( 8x + 4h-4 \)

Answer:

s:

• \( f(x + h)=\boldsymbol{4x^{2}+8xh + 4h^{2}-4x-4h} \)
• \( \frac{f(x + h)-f(x)}{h}=\boldsymbol{8x + 4h-4} \)