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.
$\text{ƒ}\left(x\right)=4x+11$

Verified step by step guidance

1. Start by finding \( f(x+h) \) by substituting \( x+h \) into the function \( f(x) = 4x + 11 \). This means replacing every \( x \) in the function with \( x+h \), so write \( f(x+h) = 4(x+h) + 11 \).
2. Next, simplify the expression for \( f(x+h) \) by distributing the 4 across \( (x+h) \), which gives \( 4x + 4h + 11 \).
3. Now, find \( f(x+h) - f(x) \) by subtracting the original function \( f(x) = 4x + 11 \) from the expression you found for \( f(x+h) \). Write this as \( (4x + 4h + 11) - (4x + 11) \).
4. Simplify the difference \( f(x+h) - f(x) \) by canceling out like terms. The \( 4x \) and \( 11 \) terms will cancel, leaving you with \( 4h \).
5. Finally, find the expression \( \frac{f(x+h) - f(x)}{h} \) by dividing the simplified difference \( 4h \) by \( h \). This will simplify to \( \frac{4h}{h} \), which you can simplify further by canceling \( h \) (assuming \( h \neq 0 \)).

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.

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 step toward finding rates 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 foundational concept in calculus, representing the slope of the secant line between two points on the function's graph.