Question

find and simplify the difference quotient \\(\frac{f(x + h) - f(x)}{h}\\) for the function \\(f(x) = 3x - 2\\).

\\(\frac{f(x + h) - f(x)}{h} = \\)

Explanation:

Evaluate the function at x + h

We substitute \(x + h\) into the given function \(f(x) = 3x - 2\).

$$ f(x + h) = 3(x + h) - 2 $$

Expanding this expression gives:

$$ f(x + h) = 3x + 3h - 2 $$

Set up the difference quotient

We substitute \(f(x + h)\) and \(f(x)\) into the difference quotient formula.

$$ \frac{f(x + h) - f(x)}{h} = \frac{(3x + 3h - 2) - (3x - 2)}{h} $$

Simplify the numerator

We distribute the negative sign and combine like terms in the numerator.

$$ (3x + 3h - 2) - (3x - 2) = 3x + 3h - 2 - 3x + 2 $$

Simplifying this yields:

$$ 3h $$

Divide by h to find the final value

We substitute the simplified numerator back into the quotient and divide.

$$ \frac{3h}{h} = 3 \quad (h eq 0) $$

Answer:

Find and simplify the difference quotient \(\frac{f(x + h) - f(x)}{h}\) for the function \(f(x) = 3x - 2\).

\(\frac{f(x + h) - f(x)}{h} =\) <blank>3</blank>