For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h. ƒ(x)=x²

Verified step by step guidance

1. Start by finding \( f(x+h) \) for the function \( f(x) = x^2 \). This means you replace every \( x \) in the function with \( x+h \), so write \( f(x+h) = (x+h)^2 \).
2. Next, expand the expression \( (x+h)^2 \) using the binomial formula \( (a+b)^2 = a^2 + 2ab + b^2 \). So, \( (x+h)^2 = x^2 + 2xh + h^2 \).
3. Now, find \( f(x+h) - f(x) \) by subtracting \( f(x) = x^2 \) from the expanded \( f(x+h) \). This gives \( (x^2 + 2xh + h^2) - x^2 \).
4. Simplify the expression \( f(x+h) - f(x) \) by canceling out \( x^2 \), leaving you with \( 2xh + h^2 \).
5. Finally, find \( \frac{f(x+h) - f(x)}{h} \) by dividing the simplified difference \( 2xh + h^2 \) by \( h \). This results in \( \frac{2xh + h^2}{h} \).

Key Concepts

Function Notation and Evaluation

Function notation, such as ƒ(x), represents a rule that assigns each input x to an output value. 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.