Questions: For the function (f(x)=-3 x^2-2 x-4), evaluate and fully simplify (f(x+h)=) (fracf(x+h)-f(x)h=)

Solution Steps

To solve the given problem, we need to evaluate the function \( f(x) = -3x^2 - 2x - 4 \) at \( x + h \) and then find the difference quotient \(\frac{f(x+h) - f(x)}{h}\).

1. Evaluate \( f(x+h) \): Substitute \( x + h \) into the function \( f(x) \) to get \( f(x+h) \).
2. Calculate the Difference Quotient: Compute \( f(x+h) - f(x) \) and then divide the result by \( h \).

To evaluate \( f(x+h) \), we substitute \( x + h \) into the function \( f(x) = -3x^2 - 2x - 4 \):

\[ f(x+h) = -3(x+h)^2 - 2(x+h) - 4 \]

Expanding this expression, we have:

\[ f(x+h) = -3(h^2 + 2hx + x^2) - 2x - 2h - 4 = -3h^2 - 6hx - 3x^2 - 2x - 2h - 4 \]

Next, we compute the difference quotient:

\[ \frac{f(x+h) - f(x)}{h} \]

Substituting the expressions we have:

\[ f(x+h) - f(x) = (-3h^2 - 6hx - 3x^2 - 2x - 2h - 4) - (-3x^2 - 2x - 4) \]

This simplifies to:

\[ f(x+h) - f(x) = -3h^2 - 6hx - 2h \]

Now, dividing by \( h \):

\[ \frac{f(x+h) - f(x)}{h} = \frac{-3h^2 - 6hx - 2h}{h} = -3h - 6x - 2 \]

Final Answer