Linear Functions

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:  

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.  

Solution:  

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

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.  

Determine the cost to make any number of bracelets.  

Solution:  

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. 

Solution:  

Step 1:  

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.? 

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 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? 

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.

• For a function f(x)=ax+b, f(0)=3 and f(1)=4. Determine the coefficients that satisfy the equation.

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.