For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h. ƒ(x)=-2x+5

Verified step by step guidance

1. Start by identifying the given function: \(f(x) = -2x + 5\).
2. To find \(f(x+h)\), substitute \(x+h\) into the function in place of \(x\). This means replacing every \(x\) in the function with \((x+h)\), so write \(f(x+h) = -2(x+h) + 5\).
3. Next, find \(f(x+h) - f(x)\) by subtracting the original function \(f(x)\) from the expression you found for \(f(x+h)\). This gives \(f(x+h) - f(x) = [-2(x+h) + 5] - [-2x + 5]\).
4. Simplify the expression \(f(x+h) - f(x)\) by distributing and combining like terms carefully.
5. Finally, to find \(\frac{f(x+h) - f(x)}{h}\), divide the simplified expression from the previous step by \(h\). This will give you the difference quotient, which is a fundamental concept in calculus and algebra for understanding rates of change.

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 the 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 represents the average rate of change of the function over the interval from x to x+h. It is a key concept in calculus, used to approximate derivatives, and helps understand the function's behavior as h approaches zero.