Need Help?

Get in touch with us

The component learnSearchBar has not been created yet.

bannerAd

Linear Functions

Sep 15, 2022
link

Key Concepts

  • Functions in function notation.
  • Linear function rule.
  • Graph of a linear function.
  • Solve problems related to linear function.

Functions, domain & range 

Function

A relation where every input has a single output.  

Domain and range are important values that help to define a relation. The domain takes all the possible input values from the setoff real numbers and the range takes all the output values of the function.  

Question 1: For the set of ordered pairs shown, identify the domain and range. Does the relation represent a function?  

{(1, 8), (5, 3), (7, 6),(2, 2), (8, 4), (3, 9), (5, 7)}{1, 8, 5, 3, 7, 6,2, 2, 8, 4, 3, 9, (5, 7)}

Solution:  

parallel

Given ordered pair: {(1, 8), (5, 3), (7, 6),(2, 2), (8, 4), (3, 9), (5, 7)}{1, 8, 5, 3, 7, 6,2, 2, 8, 4, 3, 9, (5, 7)}

Domain: {1, 5, 7, 2, 8, 3, 5}{1, 5, 7, 2, 8, 3, 5}

Range: {8, 3, 6, 2, 4, 9, 7}{8, 3, 6, 2, 4, 9, 7}

The relation represents a function because every input has a different output.  

Question 2: The flowchart shows the steps of a math puzzle. Record and fill the result in the tabular column. Make a prediction about what the final number will be for any number. Explain.  

parallel
Flowchart
Table

Solution:  

From the flowchart think of numbers 1, 2, 3, 4, and 5 and fill the table given below: 

Table 2

From the table each output is a multiple of 2.  

So, for any number/value of x, the output will be 2 times the input value.  

Function notation

Function notation is a method for writing variables as a function of other variables. 

The variable y, becomes a function of x. The variable x is used to find the value of y.  

Function notation helps to distinguish between the different functions.  

Function notation can use letters other than f. Other commonly used letters are g and h. 

Consider the equation y = 3x−2.  

Write the equation y = 3x−2

using function notation. 

The function f is defined in function notation by  f(x) = 3x−2

Example 1:  

What is the value of h(x) = 7x+1 when x = 5? 

Solution:  

Evaluate h(x) = 7x+1 for x = 5

If h(x) = 7x+1, then h(5) = 36

Example 2:  

What is the value of g(x) = 5−3x when x = 2?  

Solution:  

Evaluate g(x )= 5−3x for x = 2.  

If g(x) = 5−3x, then g(2) = −1

Linear function rule

Example 1:  

The cost to make 4 bracelets is shown in the table.  

Linear function rule:   

Determine the cost to make any number of bracelets.  

Solution:  

Step 1: 

Step 1: 

The relationship is linear.  

Step 2:  

Write a function using slope-intercept form for the rule. 

f(x) = mx+b

f(x) = 15x+b

Step 3:  

Find the value of b

Substitute any ordered pair from the table. 

17 = 15(1)+b

2=b

So, the function is

f(x) = 15x+2

Example 2:  

Write a linear function for the data in the table using function notation. 

example 2

Solution:  

Step 1:  

solution

The relationship is linear. 

Step 2:  

Write a function using the slope-intercept form for the rule. 

f(x) = mx+b

f(x) = 4x+b

Step 3:  

Find the value of b by substituting any ordered pair from the table. 

15 = 4(3)+b

3 = b

So, the linear function is

f(x) = 4x+3

Analyze a linear function  

Question: 

Tamika records the outside temperature at 6:00 A.M. The outside temperature increases by 2°F every hour for the next 6 hours. If the temperature continues to increase at the same rate, what will the temperature be at 2:00 P.M.? 

Question

Solution:  

Step 1:  

Write a function that models the situation:  

f(x) = mx+c

Since the temperature at 6:00 A.M. is −3

and the temperature is increasing at the rate of 2°F every hour.  

So,

c = −3 & m = 2

f(x) = 2x−3

Step 2:  

Sketch the graph of the function.

Step 2

Step 3:  

Find the value of y when x = 8.  

y = 2(8)−3 = 13

Given that the temperature continues to increase at the same rate, the temperature at 2:00 p.m. will be 13°F.  

The graph of f(x) = 2x−3 is a line.  

Use linear functions to solve problems 

Question: 

A chairlift starts 0.5 mi above the base of the mountain and travels up the mountain at a constant speed. How far from the base of the mountain is the chairlift after 10 minutes? 

Use linear functions to solve problems:   

Solution: 

Step 1:  

Linear function to represent the distance, the chairlift travels from the base of the mountain.  

Let the time (in minutes) be t 

Given that the speed of the chairlift is in miles per hour, convert the speed to miles per minute.  

6 miles / hour×1 hour / 60 minutes= 0.1 mile / minute

Distance traveled = rate of the chairlift × time traveling + distance from the base Distance traveled = rate of the chairlift × time traveling + distance from the base 

d(t) = 0.1×t+0.5

d(t) = 0.1t+0.5

Step 2:  

The distance of the chairlift from the base of the mountain at any time is represented by the linear function,

d(t) = 0.1t+0.5

Now, evaluate the function for

t = 1

d(10) = 0.1×10+0.5

= 1+0.5

= 1.5

So, after 10 minutes, the chairlift will be 1.5 miles up the mountainside.  

Exercise

  • What is the value of f(x)=-2x-5 when x=0, x=1, x=2, & x=-1?
  • Write a linear equation for the data in the table given below using function notation.
Exercise:
  • For a function f(x)=ax+b, f(0)=3 and f(1)=4. Determine the coefficients that satisfy the equation.

Concept Summary   

Concept Summary   

What we have learned

  • Evaluate functions in function notation.
  • Write a linear function rule.
  • Analyze a linear function.
  • Use linear functions to solve problems.

Comments:

Related topics

Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […]

Read More >>
special right triangles_01

Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem?  Right Angle Triangles A triangle with a ninety-degree […]

Read More >>
simplify algebraic expressions

Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]

Read More >>
solve right triangles

How to Solve Right Triangles?

In this article, we’ll learn about how to Solve Right Triangles. But first, learn about the Triangles.  Triangles are made up of three line segments. These three segments meet to form three angles. The lengths of the sides and sizes of the angles are related to one another. If you know the size (length) of […]

Read More >>

Other topics