Problem 55 Given the function find \(f(x+h)... [FREE SOLUTION]

Chapter 2: Problem 55

Given the function find $f(x+h)$. $$f(x)=3 x-1$$

Short Answer

Expert verified

$f(x+h) = 3x + 3h - 1$

Step by step solution

Understanding the Function

We are given the function $f(x) = 3x - 1$. This is a linear function where the coefficient of $x$ is 3, and there is a constant term, -1.

Substituting $x + h$

To find $f(x + h)$, we substitute $x + h$ in place of $x$ in the original function. So, $f(x+h) = 3(x+h) - 1$.

Distribute the 3

Apply the distributive property to distribute the 3 over $x + h$. So, this becomes $f(x+h) = 3 \cdot x + 3 \cdot h - 1$.

Simplify the Expression

Combine like terms to simplify the expression: $f(x+h) = 3x + 3h - 1$.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Function

A linear function is a basic yet vital concept in mathematics, frequently appearing in various subjects. A linear function is generally expressed in the form $ f(x) = mx + b $, where $ m $ is the slope indicating the steepness and direction of the line, and $ b $ is the y-intercept, where the line crosses the y-axis. For the given function $ f(x) = 3x - 1 $:

The slope $ m $ is 3, meaning for every 1 unit increase in $ x $, $ f(x) $ increases by 3 units.
The y-intercept $ b $ is -1, indicating the line crosses the y-axis at $ y = -1 $.

Linear functions produce a straight line graph and represent constant rate of change. They are fundamental in understanding and modeling relationships between two variables. By recognizing these patterns, you can solve real-world problems efficiently.

Distributive Property

The distributive property is a powerful algebraic tool used to simplify expressions and solve equations. It states that for any numbers or expressions $ a $, $ b $, and $ c $, $ a(b + c) = ab + ac $. This property allows us to eliminate parentheses by distributing a factor across terms within a parenthesis.
For instance, in the expression $ 3(x + h) $, distribute the 3 to both $ x $ and $ h $:

First, multiply 3 by $ x $, giving $ 3x $.
Next, multiply 3 by $ h $, giving $ 3h $.

The result is $ 3x + 3h $, showing how the distributive property helps in breaking down complex expressions into simpler parts. This technique is especially handy when dealing with algebraic expressions and equations.

Substitution

Substitution in mathematics is a useful method where you replace a variable within an expression or equation with another expression or value. In this problem, we want to find $ f(x+h) $. This means substituting $ x+h $ in place of $ x $ in the function.
Here's how to do it step-by-step:

Original function: $ f(x) = 3x - 1 $.
Substitute $ x+h $ for every $ x $: $ f(x+h) = 3(x+h) - 1 $.

By substituting, we modify the function to reflect changes in variables effectively. This technique allows you to evaluate the function's behavior with different inputs, bringing more flexibility and understanding in solving mathematical challenges.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Given the function find \(f(x+h)\). $$f(x)=3 x-1$$

Short Answer

Step by step solution

Understanding the Function

Substituting \(x + h\)

Distribute the 3

Simplify the Expression

Key Concepts

Linear Function

Distributive Property

Substitution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Mechanics Maths

Statistics

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.