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Chapter 2: Problem 63

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

Short Answer

Expert verified
22

Step by step solution

01

Identify the value of f(4)

The function \( f \) is given as a set of ordered pairs. To find \( f(4) \), locate the pair where the first element is 4. In this case, \( (4, 5) \) is the pair, so \( f(4) = 5 \).
02

Calculate the value of g(4)

The function \( g(x) = 3x + 5 \). Substitute \( x = 4 \) to find \( g(4) \). Thus, we have: \[ g(4) = 3(4) + 5 = 12 + 5 = 17. \]
03

Add the values of f(4) and g(4)

Now that both values have been determined, add them together: \[ f(4) + g(4) = 5 + 17 = 22. \]

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Key Concepts

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

Function Evaluation
In mathematics, evaluating a function means finding the result of a function for a particular value of its input.
For instance, if we have a function defined as \(\function\) and we need to find \(\function\) for \(\funcvalue\), we substitute \(\funcvalue\) into the function.
As per our example exercise, for the function \(g(x) = 3x + 5\) when \(x = 4\), we substitute to get \(g(4) = 3(4) + 5 = 17\).
Essentially, it's all about replacing the variable and simplifying.
This simple substitution approach is a fundamental skill in algebra and calculus.
Ordered Pairs
Ordered pairs are fundamental elements in the study of functions. They are written as \((x, y)\), where \(x\) is the input and \(y\) is the output.
The first number in an ordered pair is typically called the domain, and the second number is the range. For example, in the set \(\{(2,6),(3,8),(4,5)\}\), each pair represents an input-output relation.
To find \(f(4)\) from the set, locate the pair where the first number is 4. Hence, in this example, \((4, 5)\) shows that \(f(4) = 5\).
Ordered pairs are often used in graphing functions and understanding relationships between two varying quantities.
Linear Functions
A linear function can be expressed in the form \(g(x) = mx + b\), where \(m\) and \(b\) are constants.
Here, \(m\) represents the slope of the line, and \(b\) represents the y-intercept, which is where the line crosses the y-axis.
In our exercise, the linear function is given as \(g(x) = 3x + 5\).
This tells us the slope \(m\) is 3, meaning for each increase in \(x\), \(y\) increases by 3 times that amount. The y-intercept \(b\) is 5, so the graph will cross the y-axis at point (0, 5).
Linear functions are straightforward to work with because their graph is always a straight line.
They help model many real-world situations where there is a constant rate of change.

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Most popular questions from this chapter

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=\left(x^{2}-2\right)^{3}$$

Find the slope of the line \(3 x-9 y=7\).

Write a formula that expresses the diagonal \(d\) of a square as a function of the area \(A\)

Use transformations to graph each function and state the domain and range. $$y=3|x-2|$$

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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