## 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.

#### Related topics

#### Addition and Multiplication Using Counters & Bar-Diagrams

Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […]

Read More >>#### Dilation: Definitions, Characteristics, and Similarities

Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […]

Read More >>#### How to Write and Interpret Numerical Expressions?

Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division A → Addition S → Subtraction Some examples […]

Read More >>#### System of Linear Inequalities and Equations

Introduction: Systems of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities in the same variables. The following example illustrates this, y < x + 2…………..Inequality 1 y ≥ 2x − 1…………Inequality 2 Solution of a System of Linear Inequalities: A solution of a system of linear inequalities […]

Read More >>
Comments: