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Chapter 8: Problem 74

Find \(g(w)\) and \(g(w+1) .\) See Example 4. $$ g(x)=2 x-7 $$

Short Answer

Expert verified
\(g(w) = 2w - 7\); \(g(w+1) = 2w - 5\).

Step by step solution

01

Identify the function

The given function is provided as \(g(x) = 2x - 7\). We will use this function to find the outputs for specific input values.
02

Substitute and solve for g(w)

To find \(g(w)\), substitute \(w\) in place of \(x\) in the function.\[g(w) = 2w - 7\]No further simplification is possible, so this is the expression for \(g(w)\).
03

Substitute and solve for g(w+1)

To find \(g(w+1)\), substitute \(w+1\) in place of \(x\) in the function.\[g(w+1) = 2(w + 1) - 7\]Simplify the expression by distributing the 2 and combining like terms:\[g(w+1) = 2w + 2 - 7 = 2w - 5\]So, the expression for \(g(w+1)\) is \(2w - 5\).

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Key Concepts

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

Substitution Method
The substitution method is a fundamental technique in algebra used to evaluate expressions or solve equations. It's like plugging in a number or expression wherever you see a certain variable. In our exercise, the function provided is \( g(x) = 2x - 7 \). We needed to find what happens when \( x \) is substituted by other expressions like \( w \) and \( w+1 \). This method helps simplify expressions by replacing variables with specific numbers or other expressions.

Here鈥檚 what we did:
  • Replace \( x \) with \( w \) to find \( g(w) \). This substitution results in the expression \( g(w) = 2w - 7 \).
  • For \( g(w+1) \), substitute \( w+1 \) for \( x \). This gives the expression \( g(w+1) = 2(w+1) - 7 \).
By using substitution, we effectively determine how the function behaves for different inputs.
Function Evaluation
Function evaluation is the process of determining the output of a function for a given input. It's crucial for understanding how the function works and how it produces results based on input values.

In this case, we looked at how the function \( g(x) = 2x - 7 \) behaves when different expressions are used in place of \( x \):
  • Firstly, \( g(w) \) is found by plugging \( w \) into the function: \( g(w) = 2w - 7 \).
  • Secondly, \( g(w+1) \) involves substituting \( w+1 \) into the function, leading to \( 2(w+1) - 7 \).
This function evaluation is key in understanding how variations in the input affect the function鈥檚 output. It allows us to predict and calculate every possible outcome given different inputs.
Algebraic Simplification
Algebraic simplification is the process of making an expression more straightforward or easier to understand. When you have complex expressions, simplifying them can help you see patterns or solutions more clearly.

In the exercise, once we performed the substitution for \( g(w+1) \), we were left with the expression:
  • \( g(w+1) = 2(w + 1) - 7 \)
  • Distribute the 2: \( 2 \times (w+1) = 2w + 2 \)
  • Combine like terms: \( 2w + 2 - 7 = 2w - 5 \)
By combining like terms, we've reached the simplest form for \( g(w+1) \), which is \( 2w - 5 \). Simplifying expressions helps in clearly identifying the relationships and operations present in the algebraic function. It's an essential skill for solving problems more efficiently.

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Most popular questions from this chapter

Graph each rational function. Show the vertical asymptote as a dashed line and label it. $$f(x)=\frac{1}{x-1}$$

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Slope \(\frac{1}{4},\) passes through \((8,1)\)

$$\text { Simplify: }\left[\left(x^{-1}+1\right)^{-1}+1\right]^{-1}$$

Solve each equation. See Example \(10 .\) $$\frac{7}{5 x}-\frac{1}{2}=\frac{5}{6 x}+\frac{1}{3}$$

In the following problems, simplify each expression by performing the indicated operations and solve each equation. $$\frac{y^{3}-x^{3}}{2 x^{2}+2 x y+x+y} \cdot \frac{2 x^{2}-5 x-3}{y x-3 y-x^{2}+3 x}$$
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