Skip to main content
Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 54
Chapter 3, Problem 54

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.
ƒ(x)=1+2x2ƒ(x)=1+2x^2

Verified step by step guidance
1
Start by writing down the given function: \(f(x) = 1 + 2x^2\).
To find \(f(x+h)\), substitute every \(x\) in the function with \((x+h)\), so write \(f(x+h) = 1 + 2(x+h)^2\).
Next, expand the squared term \((x+h)^2\) using the formula \((a+b)^2 = a^2 + 2ab + b^2\), which gives \(x^2 + 2xh + h^2\). Substitute this back into \(f(x+h)\).
Calculate \(f(x+h) - f(x)\) by subtracting the original function \(f(x) = 1 + 2x^2\) from the expression you found for \(f(x+h)\).
Finally, find the difference quotient by dividing the expression \(f(x+h) - f(x)\) by \(h\), which is \( rac{f(x+h) - f(x)}{h}\). Simplify this expression as much as possible.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
Recommended video:
4:26
Evaluating Composed Functions

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.
Recommended video:
4:56
Function Composition

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.
Recommended video:
3:49
Product, Quotient, and Power Rules of Logs
Related Practice
Textbook Question

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.

ƒ(x)=1−x2ƒ(x)=1-x^2

Textbook Question

Find the slope of each line, provided that it has a slope. 11x + 2y = 3

Textbook Question

Graph each function. Give the domain and range. ƒ(x)=-[[x]]

1
views
Textbook Question

Let ƒ(x)=-3x+4 and g(x)=-x2+4x+1. Find each of the following. Simplify if necessary. g(10)

3
views
Textbook Question

Graph each function. Give the domain and range. ƒ(x)=[[2x]]

Textbook Question

Let ƒ(x)=-3x+4 and g(x)=-x2+4x+1. Find each of the following. Simplify if necessary. g(-2)

1
views