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Chapter 2: Problem 58

Let \(f=\\{(2,6),(3,8),(4,5)\\}\) and \(g(x)=3 x+5 .\) Find the following. $$g(4)$$

Short Answer

Expert verified
g(4) = 17

Step by step solution

01

- Understand the Function g(x)

The function given is a linear function defined as \( g(x) = 3x + 5 \).
02

- Substitute the Value

To find \( g(4) \), substitute \( x = 4 \) into the function \( g(x) \).
03

- Perform the Calculation

Replace \( x \) with \( 4 \) in the function: \( g(4) = 3(4) + 5 \).
04

- Simplify the Expression

Calculate the expression: \( g(4) = 3(4) + 5 = 12 + 5 = 17 \).

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Key Concepts

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

Function Evaluation
Function evaluation means finding the value of a function at a particular input. For understanding, think of functions like machines that take an input and provide an output. In our example, we have the linear function \( g(x) = 3x + 5 \). To find \( g(4) \), we are essentially putting 4 into the function 'machine'.

By doing this, we follow several steps:
  • Identify the function and its components
  • Substitute the given input value
  • Perform the calculation to get the output
Through these steps, you can determine the specific output value for any given input value into the function.
Substitution
Substitution is a crucial step in function evaluation where you replace the variable with a given number. In the given problem, we replace the \( x \) in \( g(x) \) with 4. This gives us \( g(4) = 3(4) + 5 \).

Here鈥檚 a simple way to think about substitution:
  • Identify the variable in the function
  • Copy the rest of the function around the variable
  • Replace the variable with the specified number
This is an important skill not just in evaluating functions but also in solving algebraic equations and more complex mathematical problems.
Linear Equations
Understanding linear equations helps greatly in tackling functions like \( g(x) = 3x + 5 \). A linear equation is an equation of the first degree, meaning it has the variable raised to the power of one. The graph of a linear equation is a straight line.

In our example, \( g(x) = 3x + 5 \) is a linear equation in slope-intercept form: \( y = mx + b \), where
  • \( m \) represents the slope (here it is 3)
  • \( b \) represents the y-intercept (here it is 5)
This form allows easy substitution and subsequent evaluation for different values of \( x \). Recognizing when you're working with a linear equation and its properties simplifies many steps in solving problems.
Function Notation
Function notation like \( f(x) \) or \( g(x) \) is a way to name functions and specify that the function depends on \( x \). In our example, \( g(x) = 3x + 5 \) is a function that gives outputs based on the input \( x \).

The notation alerts you to the process you鈥檒l follow: using the function's rule to compute the output for a given input. Some key points of function notation are:
  • The letter (e.g., \( g \)) names the function
  • The parentheses and variable (e.g., \( (x) \)) show the input
  • The formula provides the rule for computing the output
Understanding how to read and work with function notation simplifies evaluating, transforming, and interpreting functions.

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Most popular questions from this chapter

Use transformations to graph each function and state the domain and range. $$y=-\frac{1}{2} x+40$$

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. $$y=x^{5}-13 x^{3}+36 x$$

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Find and simplify \(f(a+2)-f(a)\) given that \(f(x)=x^{2}+1\)

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=(x-1)^{2}$$
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