Chapter 1: Functions

Question 1

Week 1

What is a variable?

Answer 1

A variable is a quantity that can change e.g. time, height, age.

Question 2

Week 1

What are the different types of variables.

Answer 2

There are two types of variables

• Independent variables.
• Dependent variables.

Independent variables are variable quantities that do not depend on other variables that you are trying to measure.

Dependent variables are variable quantities that do depend on other variables that you measure. This means that if you change the value of one variable, the value that depends on the one you changed will also change.

Question 3

Week 1

What is a function?

Answer 3

If a variable y is dependent on a variable x in such a way that each value of x determines exactly one value of y, we say that y is a function of x.

Essentially a function is an operation that takes an input x, and performs a series of operations on it to output y.

Suppose that you are baking a cake. You mix all the ingredients (the input) and pop the batter into the oven (the function). The oven will then heat the batter and give you a cake (the output).

We can also infer that the output (y) is dependent on the input (x).

Question 4

Week 1

What disqualifies a relation from becoming a function?

Answer 4

An equation that maps one x value to two y values cannot be a function.

Some examples of non-functional equations include:

• The equation of a circle e.g. x² + y² = 25
• The equation y = ± x

Question 5

Week 1

What is the domain of a function?

Answer 5

The domain of a function is the set of all possible input values that the function can accept.

Question 6

What is the range of a function?

Answer 6

The range of a function is the set of all real values that are a result of the input values.

f(x) is called the range of f.

Question 7

Week 1

What is the co-domain of a function?

Answer 7

The co-domain of a function is the set of all possible output values that the function can produce regardless of the domain.

Question 8

Week 1

Describe the graphical approach to find the domain of a function.

Answer 8

1) The denominator of a fraction cannot be 0
2) The square root expression in the numerator should be greater than or equal to zero.
3) The square root expression in the denominator should be greater than zero.

Question 9

Week 1

Describe the graphical approach to find the range of a function.

Answer 9

1) Write the equation as f(x) = y
2) Express y as the function of x
3) Eliminate the values of y according to the question.
4) Find the question.

Question 10

Week 1

Describe how to graph a function.

Answer 10

1) Identify the interval to sketch.
2) Take the x values and find the corresponding y values
3) Draw the x-y plane.
4) Calculate the x and y intercepts.
5) Mark the ordered pair in the x-y plane and sketch the graph.

Question 11

Week 1

How do you test for a function?

Answer 11

The Vertical Line Test

• The vertical line test is a visual way to determine if a graph is a function or not.
• A function can only have one unique output. No vertical line can intersect the graph of a function more than once.
• If it does, the graph is not a function.

Question 12

Describe a piece-wise defined function.

Answer 12

When a function is defined in pieces by using different formulae on different parts of the domain is called a piece-wise function.

In a piece-wise function, the function changes based on the input value.

For example, different cake batters require different baking temperatures, so if you bake a red velvet cake instead of a vanilla cake, you will have to adjust the baking time accordingly in order to obtain a well-cooked cake.

Similarly, a piece-wise function can change itself based on the inputs it receives in order to output a coherent function.

An example of a function that is defined in pieces is the absolute value function.

Question 13

Week 1

What is the Greatest Integer Function (GIF)?

Answer 13

The greatest integer function (also known as the integer floor function) is denoted using the following formula

f(x) = [[x]]

What the GIF does is that it rounds whatever x value it receives down to the nearest integer. For example

f(5.99) = [[5.99]] = 5

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To learn more about the GIF and its graph, visit this link.

Question 14

Week 1

What is the Least Integer Function (LIF)?

Answer 14

The least integer function (also known as the integer ceiling function) is denoted using the following formula

f(x) = [x]

What the LIF does is that it rounds whatever x value it receives up to the nearest integer. For example

f(5.01) = [[5.01]] = 6

To learn more about the LIF and its graph, visit this link.

Question 15

Week 2

What is an even function?

Answer 15

The graph of an even function is symmetric to the y-axis.

If (x,y) exists, (-x,y) also exists.

Some examples of even functions include:

f(x) = x²
f(x)= x² + 1

Question 16

Week 2

What is an odd function?

Answer 16

The graph of an odd function is symmetric about the origin.

If (x,y) exists, (-x,-y) also exists.

Some examples of odd functions include:

f(x) = x³
f(x) = x⁶

Question 17

Week 2

Describe the different operations performed using functions i.e. addition, subtraction, multiplication, division.

Answer 17

Let’s assume you have two functions f(x) and g(x)

f(x) = x²
g(x) = x

Only four operations can be performed using these functions, which are:

• Addition ( f(x) + g(x) = x² + x )
• Subtraction ( f(x) - g(x) = x² - x )
• Multiplication ( f(x) * g(x) = x³ )
• Division ( f(x) / g(x) = x )

The domain of the resultant function is the intersection of the domain of f and the domain of g.

Question 18

Week 2

What is a composite function?

Answer 18

A composite function such as fog (x) (also known as f(g(x))) is a function of f that whose x value is another function g(x) in itself.

What is the domain of f(g(x))?

Let’s assume that f(x) = x² + 5 and g(x) = 5x

f(g(x)) = f(5x) = (5x)² + 5 = 25x² + 5

The domain of f(g(x)) in this case is all real numbers or (-∞, +∞).

The ∞ of f(g(x)) in this case is [5, +∞).

Question 19

Week 2

How do you vertically and horizontally shift graphs?

Answer 19

The translations of graphs can happen in two ways: horizontal and vertical.

Horizontal Shift

Let’s assume that f(x) = x²

f(x+1) = (x+1)² (horizontal shift 1 unit to the left.

f(x-1) = (x -1)² (horizontal shift 1 unit to the right.

From this example, we can infer that a graph is shifted horizontally only when the value of x is changed.

Vertical Shift

Let’s assume that f(x) = x²

f(x) + 1 = x² + 1 (vertical shift 1 unit up)

f(x) -1 = x² - 1 (vertical shift 1 unit down)

From this example, we can infer that a graph is shifted horizontally only when the value of f(x) is changed.

Question 20

Week 2

Explain the concept of stretching and compressing a graph.

Answer 20

The scaling of graphs can be performed in two ways: horizontal and vertical.

Horizontal Scaling

Let’s assume that f(x) = x²

f(2x) = (2x)² (horizontal compression by a factor of 2)

f(x/2) = (x/2)² (horizontal stretch by a factor of 2)

From this example, we can infer that a graph is scaled horizontally only when the value of x is scaled.

Vertical Scaling

Let’s assume that f(x) = x²

2 * f(x) = 2(x²) (vertical stretch by a factor of 2)

1/2 * f(x) = 1/2(x²) (vertical compression by a factor of 2)

From this example, we can infer that a graph is shifted horizontally only when the value of f(x) is scaled.

Question 21

Week 2

Explain the concept of reflecting a graph.

Answer 21

If the graph of f(x) is multiplied with a scale factor c such that c = -1, one of two things can happen:

• Reflection across the x-axis (-f(x))
• Reflection across the y-axis (f(-x))