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Chapter 2: Problem 67

Let \(f(x)=3 x^{2}-x\) and \(g(x)=4 x-2 .\) Find the following. $$g(a+2)$$

Short Answer

Expert verified
g(a+2) = 4a + 6

Step by step solution

01

- Understand the Function g(x)

Recall that the function given is \( g(x) = 4x - 2 \). We need to evaluate this function at a specific input.
02

- Substitute the Expression

Substitute \( a+2 \) into the function \( g(x) \): \( g(a+2) = 4(a+2) - 2 \).
03

- Simplify the Expression

Expand and simplify the expression: \( 4(a+2) - 2 = 4a + 8 - 2 \).
04

- Combine Like Terms

Combine the constants: \( 4a + 8 - 2 \) simplifies to \( 4a + 6 \).

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Key Concepts

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

substitution
Substitution is a fundamental concept in algebra and mathematics as a whole. It involves replacing a variable or an expression with a given value or another expression. Here, let's consider the function
g(x) = 4x - 2.
We are tasked with evaluating this function at a + 2.
To achieve this, we substitute a + 2 for x in the function. This means everywhere we see x in the function, we replace it with a + 2.
After substitution, our expression becomes:
g(a + 2) = 4(a + 2) - 2.
Substitution helps to simplify our calculations by making it clear what we are working with. Always keep track of what you substitute to avoid confusion.
simplification
Simplification is the process of making an algebraic expression as simple as possible. Here, after substituting a + 2 into the function g(x), we need to simplify the expression:
g(a + 2) = 4(a + 2) - 2.
First, we distribute the 4 through the parentheses:
= 4 * a + 4 * 2 - 2.
This gives us:
= 4a + 8 - 2.
Simplifying an expression helps to clarify and reduce it to fewer terms or simpler forms, making it easier to work with in subsequent calculations. Taking each step slowly and ensuring each part is correct is key to successful simplification.
combining like terms
Combining like terms is an important step in simplifying algebraic expressions. Like terms are terms that have identical variable parts. In our example, 4a + 8 - 2, the like terms are the constants 8 and -2.
To combine them, we simply add or subtract the coefficients:
8 - 2 = 6.
So, the expression 4a + 8 - 2 simplifies to:
4a + 6.
Combining like terms helps to tidy up an expression, reducing it to the simplest possible form, which is crucial for solving equations efficiently. Always identify and group like terms together to keep your work organized.

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Most popular questions from this chapter

Use transformations to graph each function and state the domain and range. $$y=-\sqrt{x+2}-4$$

Make a table listing ordered pairs for each function. Then sketch the graph and state the domain and range. Identify any intervals on which \(f\) is increasing, decreasing, or constant. $$f(x)=|x-1|$$

Solve \(|13 x-55|=9\).

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