Question

1. find the simplified difference quotient for the function ( f(x) = x^2 + 2x - 7 )

Explanation:

Step1: Recall the difference quotient formula

The difference quotient of a function \( f(x) \) is given by \( \frac{f(x + h)-f(x)}{h} \), where \( h
eq0 \).

Step2: Find \( f(x + h) \)

Given \( f(x)=x^{2}+2x - 7 \), substitute \( x\) with \( x + h \):
\( f(x + h)=(x + h)^{2}+2(x + h)-7 \)
Expand \( (x + h)^{2} \) using the formula \( (a + b)^{2}=a^{2}+2ab + b^{2} \), here \( a = x \) and \( b = h \), so \( (x + h)^{2}=x^{2}+2xh+h^{2} \).
Then \( f(x + h)=x^{2}+2xh+h^{2}+2x + 2h-7 \)

Step3: Compute \( f(x + h)-f(x) \)

Subtract \( f(x)=x^{2}+2x - 7 \) from \( f(x + h) \):
\( f(x + h)-f(x)=(x^{2}+2xh+h^{2}+2x + 2h-7)-(x^{2}+2x - 7) \)
Remove the parentheses: \( x^{2}+2xh+h^{2}+2x + 2h-7 - x^{2}-2x + 7 \)
Simplify by combining like terms: \( x^{2}-x^{2}+2x-2x+2xh+h^{2}+2h - 7 + 7=2xh+h^{2}+2h \)

Step4: Divide by \( h \) to get the difference quotient

\( \frac{f(x + h)-f(x)}{h}=\frac{2xh+h^{2}+2h}{h} \)
Factor out \( h \) from the numerator: \( \frac{h(2x + h+2)}{h} \)
Since \( h
eq0 \), we can cancel out \( h \) from the numerator and the denominator: \( 2x+h + 2 \)

Answer:

The simplified difference quotient is \( 2x + h+2 \)