Question
Mathematics
If $f\left(x\right)=-4{x}^{2}$f(x)=-4x^(2), simplify: $\frac{f\left(-2+h\right)-f\left(-2\right)}{h}$(f(-2+h)-f(-2))/(h). Show all steps neatly.
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We are given the function $f\left(x\right)=-4{x}^{2}$f(x)=-4x^(2) and we need to simplify the expression $\frac{f\left(-2+h\right)-f\left(-2\right)}{h}$(f(-2+h)-f(-2))/(h).
Substituting $x=-2$x=-2 into the equation for $f\left(x\right)$f(x), we get:
$f\left(-2\right)=-4\left(-2{\right)}^{2}=-16$f(-2)=-4(-2)^(2)=-16
Substituting $x=-2+h$x=-2+h into the equation for $f\left(x\right)$f(x), we get:
$f\left(-2+h\right)=-4\left(-2+h{\right)}^{2}=-4\left({h}^{2}-4h+4\right)=-4{h}^{2}+16h-16$f(-2+h)=-4(-2+h)^(2)=-4(h^(2)-4h+4)=-4h^(2)+16 h-16
Now, we can substitute these values into the expression we need to simplify:
$\frac{f\left(-2+h\right)-f\left(-2\right)}{h}=\frac{\left(-4{h}^{2}+16h-16\right)-\left(-16\right)}{h}$(f(-2+h)-f(-2))/(h)=((-4h^(2)+16 h-16)-(-16))/(h)
Simplifying the numerator, we get:
$\frac{-4{h}^{2}+16h-16+16}{h}=\frac{-4{h}^{2}+16h}{h}$(-4h^(2)+16 h-16+16)/(h)=(-4h^(2)+16 h)/(h)
Factoring out $4h$4h from the numerator, we get:
$\frac{4h\left(-h+4\right)}{h}$(4h(-h+4))/(h)
Canceling the factor of $h$h in the numerator and denominator, we get:
$4\left(-h+4\right)=\overline{)-4h+16}$4(-h+4)=-4h+16
Therefore, the simplified expression is $-4h+16$-4h+16.
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