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Chapter 2: Problem 74

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

Short Answer

Expert verified
\( g(x+2) - g(x) = 8 \)

Step by step solution

01

Understand the Problem

We need to evaluate the expression \(g(x+2) - g(x)\) given the function \(g(x) = 4x - 2\).
02

Substitute \(x + 2\) into \(g(x)\)

Calculate \(g(x+2)\) by substituting \(x+2\) into the function \(g(x)\). \[g(x+2) = 4(x+2)-2\]
03

Simplify \(g(x+2)\)

Simplify the expression for \(g(x+2)\). \[g(x+2) = 4(x+2)-2 = 4x + 8 - 2 = 4x + 6\]
04

Write the Expression for \(g(x)\)

Recall the given function \(g(x) = 4x - 2\).
05

Subtract \(g(x)\) from \(g(x+2)\)

Subtract \(g(x)\) from \(g(x+2)\) to find the desired expression. \[g(x+2) - g(x) = (4x + 6) - (4x - 2)\]
06

Simplify

Combine like terms to simplify the expression. \[g(x+2) - g(x) = 4x + 6 - 4x + 2 = 8\]

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Key Concepts

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

function substitution
Function substitution involves replacing the variable in a function with another expression. Here, we substituted the expression \(x + 2\) into the function \(g(x)\). Understanding this step is crucial, as it allows you to evaluate how changes in the input affect the output.

Let's see how we did this with \(g(x)\). The original function given is \(g(x) = 4x - 2\). When substituting \(x+2\) for \(x\), you replace every instance of \(x\) in the function with the new expression.

So, for \(g(x + 2)\):
  • Start with the function \(g(x) = 4x - 2\).
  • Replace \(x\) with \(x + 2\).
  • This gives us \(g(x + 2) = 4(x + 2) - 2\).
This substitution helps us see how functions change and adapt based on new inputs.
expression simplification
Expression simplification is about reducing complex expressions to simpler forms. After substituting \(x + 2\) into \(g(x)\), you get the expression \(4(x + 2) - 2\).

The next step was breaking this down:
  • Distribute the \(4\) to both terms inside the parentheses: \(4x + 8\).
  • Then, subtract \(2\) from \(8\), yielding: \(4x + 6\).
Each step in simplifying ensures we handle all parts of the expression correctly, making it easier to work with. Breaking it down step by step helps avoid mistakes and gives us the simplest form of the expression, making further operations straightforward.

This simplified form, \(4x + 6\), is much easier to manage in mathematical processes, such as finding differences or further substitutions.
difference of functions
The difference of functions means subtracting one function from another. This concept helps us understand how the outputs of different functions relate.

In the problem \(g(x+2) - g(x)\), after substituting and simplifying, our goal was to find out how much \(g(x)\) changes when \(x\) increases by \(2\).

Here鈥檚 how we did it:
  • We started with the expressions \(g(x+2) = 4x + 6\) and the original \(g(x) = 4x - 2\).
  • Next, subtract \(g(x)\) from \(g(x+2)\): \(4x + 6 - (4x - 2)\).
  • Notice the terms \(4x\) cancel out each other, leaving \(6 + 2\).
  • The result is \(8\).
This tells us that the function increases by \(8\) units when \(x\) increases by \(2\).

Understanding the difference of functions is vital in calculus and algebra, providing insights into function behavior and changes in varying inputs.

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Most popular questions from this chapter

Draw a graph that pictures each situation. Explain any choices that you make. Identify the independent and dependent variables. Determine the intervals on which the dependent variable is increasing, decreasing, or constant. Answers may vary. Starting from the pit, Helen made three laps around a circular race track at 40 seconds per lap. She then made a 30 -second pit stop and two and a half laps before running off the track and getting stuck in the mud for the remainder of the five-minute race. Graph Helen's distance from the pit as a function of time.

Solve \(2 w^{2}-5 w-9=0\).

Draw a graph that pictures each situation. Explain any choices that you make. Identify the independent and dependent variables. Determine the intervals on which the dependent variable is increasing, decreasing, or constant. Answers may vary. Captain Janeway left the holodeck at 7: 45 to meet Tuvok, her chief of security, on the Bridge. After walking for 3 minutes, she realized she had forgotten her tricorder and returned to get it. She picked up the tricorder and resumed her walk, arriving on the Bridge at \(8: 00 .\) After 15 minutes the discussion was over, and Janeway returned to the holodeck. Graph Janeway's distance from the holodeck as a function of time.

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)=\left\\{\begin{array}{cll} x+3 & \text { for } & x \leq-2 \\ \sqrt{4-x^{2}} & \text { for } & -2

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=x^{4}-x^{3}$$
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