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Chapter 2: Problem 71

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

Short Answer

Expert verified
g(x + h) = 4x + 4h - 2.

Step by step solution

01

- Understand the Problem

We need to find the expression for the function g when the input is substituted with \(x + h\). The function \(g(x)\) is given by \(g(x) = 4x - 2\).
02

- Substitute \(x+h\) into g(x)

Substitute \(x + h\) into the function \(g(x)\) to get \(g(x + h)\). This means we replace every \(x\) in \(g(x)\) with \(x + h\).
03

- Perform the Substitution

Substitute \(x + h\) into \(g(x) = 4x - 2\). This becomes \(g(x + h) = 4(x + h) - 2\).
04

- Simplify the Equation

Distribute the 4 in the equation: \(g(x + h) = 4x + 4h - 2\).

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Key Concepts

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

Substitution in Functions
When working with algebraic functions, one of the most important skills is substitution. In this exercise, we substitute a value or expression into a function. For example, given a function like \(g(x) = 4x - 2\), if we want to find \(g(x + h)\), we replace every occurrence of \(x\) with \(x + h\).

This process is straightforward:
  • Identify the expression to substitute.
  • Replace the variable in the function with that expression.
For our function \(g(x + h)\), we substitute \(x + h\) for \(x\), resulting in \(g(x + h) = 4(x + h) - 2\). Mastering substitution is crucial for dealing with more complex algebraic expressions and functions later on.
Simplifying Expressions
Once you perform substitution in a function, the next crucial step is simplifying the resulting expression. Simplifying helps in making the equation more manageable and easier to interpret. In our given exercise, after substituting \(x + h\) into \(g(x)\), we get \(g(x + h) = 4(x + h) - 2\).

The key steps to simplify this expression are:
  • Distribute any constants or coefficients through the expression: Distribute the 4 in the equation, which gives us \(g(x + h) = 4x + 4h - 2\).
  • Combine any like terms, if possible: In this case, there are no like terms to combine further.
Simplifying the expression makes it more concise and often reveals the function's behavior or properties more clearly.
Function Operations
Functions can often interact with one another through various operations. These include addition, subtraction, multiplication, and division of functions, as well as composition. Understanding these operations allows for greater flexibility in manipulating and interpreting functions.

Let's consider some basic operations:
  • Addition: Adding two functions \(f(x)\) and \(g(x)\) means forming the new function \(f(x) + g(x)\).
  • Subtraction: Subtracting function \(g(x)\) from \(f(x)\) results in \(f(x) - g(x)\).
  • Multiplication: Multiplying two functions means finding \(f(x) \times g(x)\).
  • Division: Dividing function \(f(x)\) by \(g(x)\), provided \(g(x) eq 0\), gives \(\frac{f(x)}{g(x)}\).
In our exercise, we performed substitution within a single function rather than composing two functions, but the understanding of these operations is essential as you progress in algebra.

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Most popular questions from this chapter

Use the minimum and maximum features of a graphing calculator to find the intervals on which each function is increasing or decreasing. Round approximate answers to two decimal places. $$y=x^{4}-11 x^{2}+18$$

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