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Chapter 2: Problem 68

Let \(f(x)=3 x^{2}-x\) and \(g(x)=4 x-2 .\) Find the following. $$g(a-5)$$

Short Answer

Expert verified
4a - 22

Step by step solution

01

Understand the function g(x)

The function given is \( g(x) = 4x - 2 \). We need to find the value of \( g \) at \( a-5 \). This means substituting \( a-5 \) in place of \( x \) in the function \( g(x) \).
02

Substitute \( a-5 \) into \( g(x) \)

Substitute \( a-5 \) for \( x \) in the function \( g(x) \). This gives: \[ g(a-5) = 4(a-5) - 2 \]
03

Simplify the expression

Distribute the 4 through the parenthesis: \[ g(a-5) = 4a - 20 - 2 \]. Combine like terms to simplify: \[ g(a-5) = 4a - 22 \]

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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 mathematics which involves replacing a variable with a given value or expression. In our exercise, we substitute the expression \(a-5\) into the function \(g(x)\). This means that, wherever we see an \(x\) in \(g(x)\), we replace it with \(a-5\). This step allows us to evaluate the function at specific points or expressions. Substitution is key in helping us understand how different values influence the function.
simplification
Simplification is the process of combining like terms and performing arithmetic operations to condense an expression or equation into its simplest form. After substituting \(a-5\) into the function \(g(x)\), we have \(g(a-5) = 4(a-5) - 2\). To simplify this, we distribute the 4 through the parentheses to get \(4a - 20\) and then subtract 2. Combining these terms, we get \(4a - 22\), which is a more concise and manageable form of the expression. Simplification makes expressions easier to work with and understand.
polynomial functions
Polynomial functions are expressions that include coefficients and variables raised to non-negative integer powers. The functions \(f(x) = 3x^2 - x\) and \(g(x) = 4x - 2\) are examples of polynomial functions. These functions play a crucial role in algebra and calculus due to their straightforward nature and the patterns they reveal. Understanding how to manipulate and evaluate polynomial functions, like substituting variables and simplifying expressions, is essential for solving various mathematical problems.
variable expression
A variable expression includes numbers, variables, and arithmetic operations, but it doesn't have an equal sign like an equation. In our example, the expression \(a-5\) is used as a variable expression. Evaluating this within the function \(g(x)\) requires substituting it into the function and simplifying. By mastering variable expressions, you can manipulate and simplify more complex mathematical relationships, making it easier to solve real-world problems.

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Most popular questions from this chapter

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