For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4.
$\text{ƒ}\left(x\right)=x^2-4x+2$

Verified step by step guidance

1. Start by finding \( f(x+h) \) by substituting \( x+h \) into the function \( f(x) = x^2 - 4x + 2 \). This means replacing every \( x \) in the expression with \( x+h \), so write \( f(x+h) = (x+h)^2 - 4(x+h) + 2 \).
2. Next, expand the squared term and distribute the \( -4 \) across \( (x+h) \). Recall that \( (x+h)^2 = x^2 + 2xh + h^2 \) and \( -4(x+h) = -4x - 4h \). So, rewrite \( f(x+h) \) as \( x^2 + 2xh + h^2 - 4x - 4h + 2 \).
3. Now, find \( f(x+h) - f(x) \) by subtracting the original function \( f(x) = x^2 - 4x + 2 \) from the expression you found for \( f(x+h) \). Write this as \( (x^2 + 2xh + h^2 - 4x - 4h + 2) - (x^2 - 4x + 2) \).
4. Simplify the expression \( f(x+h) - f(x) \) by canceling out like terms. Notice that \( x^2 \), \( -4x \), and \( +2 \) will cancel out, leaving you with the terms involving \( h \).
5. Finally, find \( \frac{f(x+h) - f(x)}{h} \) by dividing the simplified expression from the previous step by \( h \). This will give you the difference quotient, which is a key concept in understanding rates of change and the derivative.

Key Concepts

Function Notation and Evaluation

Function notation, such as ƒ(x), represents a rule that assigns each input x to an output. Evaluating ƒ(x+h) means substituting x+h into the function in place of x, which helps analyze how the function behaves when its input changes by h.

Difference of Function Values

The expression ƒ(x+h) - ƒ(x) calculates the change in the function's output as the input changes from x to x+h. This difference is fundamental in understanding how the function varies over an interval and is a stepping stone toward concepts like average rate of change.

Difference Quotient

The difference quotient, [ƒ(x+h) - ƒ(x)] / h, measures the average rate of change of the function over the interval from x to x+h. It is a key concept in calculus, representing the slope of the secant line, and is used to approximate derivatives as h approaches zero.