Unit notes and practice questions

Grade 11 Mathematics unit 1: Relations and Functions

Everything here is for unit 1 of the Grade 11 Mathematics textbook: what the chapter covers, the short notes, practice questions with their answers, and the lesson video where one exists.

8
Cards: 10
Practice questions on unit 1: 1–82
The textbook pages for unit 1

What unit 1 covers

1.Relations
2.Inverse of Relations and Their Graphs
3.Types of Functions
4.Composition of Functions
5.Inverse Functions and their Graphs
6.Applications of Relations and Functions

Short notes for unit 1

The cards filed under this chapter on the Grade 11 Mathematics page. Open one at a time.

Absolute Value FunctionFormula

|a| = \begin{cases} a & \text{if } a > 0 \\ 0 & \text{if } a = 0 \\ -a & \text{if } a < 0 \end{cases}

|a| = \begin{cases} a & \text{if } a > 0 \\ 0 & \text{if } a = 0 \\ -a & \text{if } a < 0 \end{cases}

$a$: real number $dimensionless$
$|a|$: absolute value or modulus of a $dimensionless$

When you use it

Use this definition to evaluate the modulus of any real number or to simplify expressions inside absolute value bars.

Watch out

The absolute value of any real number is always nonnegative.

Drafted from Grade 11 Mathematics, pages 1-82, then checked twice before it went up

Domain and Range of a RelationRule

When you use it

Use this to identify the domain and range from a relation given as a set of ordered pairs.

Watch out

Domain contains only the first coordinates and range contains only the second coordinates. Do not write duplicate elements in either set.

Drafted from Grade 11 Mathematics, pages 1-82, then checked twice before it went up

Equality of Ordered PairsCommon mistake

When you use it

Use this when checking if two ordered pairs are equal or when finding unknown coordinates.

Watch out

In an ordered pair (x, y), order matters. The pair (x, y) equals (y, x) only when x equals y.

Drafted from Grade 11 Mathematics, pages 1-82, then checked twice before it went up

Graph of an Inverse FunctionMethod

When you use it

Use this to sketch the graph of an inverse function from the graph of the original function.

Watch out

Reflect every point across the line y = x by interchanging the x and y coordinates.

Drafted from Grade 11 Mathematics, pages 1-82, then checked twice before it went up

Inverse Function ConditionRule

When you use it

Use this to verify whether two given functions f and g are inverses of each other.

Watch out

You must check both compositions. Showing only one direction is not sufficient.

Drafted from Grade 11 Mathematics, pages 1-82, then checked twice before it went up

Inverse Function NotationCommon mistake

When you use it

Use this when interpreting the notation for the inverse of a function.

Watch out

The inverse function f^{-1}(x) is not equal to -f(x) and is not equal to 1/f(x).

Drafted from Grade 11 Mathematics, pages 1-82, then checked twice before it went up

Negative Rational Power FunctionsReference tablef(x) = x^-1/n · f(x) = x^-m/n

Function	m	n	Domain	Range
$f(x) = x^{-1/n}$	$1$	even	$(0, \infty)$	$(0, \infty)$
$f(x) = x^{-1/n}$	$1$	odd	$\mathbb{R} \setminus \{0\}$	$\mathbb{R} \setminus \{0\}$
$f(x) = x^{-m/n}$	odd	even	$(0, \infty)$	$(0, \infty)$
$f(x) = x^{-m/n}$	even	odd	$\mathbb{R} \setminus \{0\}$	$(0, \infty)$

When you use it

Use this table to determine the domain and range of power functions with negative rational exponents.

Watch out

Zero is never in the domain of a negative power function because division by zero is undefined.

Drafted from Grade 11 Mathematics, pages 1-82, then checked twice before it went up

Positive Rational Power FunctionsReference tablef(x) = x^1/n · f(x) = x^m/n

Function	m	n	Domain	Range
$f(x) = x^{1/n}$	$1$	even	$[0, \infty)$	$[0, \infty)$
$f(x) = x^{1/n}$	$1$	odd	$\mathbb{R}$	$\mathbb{R}$
$f(x) = x^{m/n}$	odd	even	$[0, \infty)$	$[0, \infty)$
$f(x) = x^{m/n}$	even	odd	$\mathbb{R}$	$[0, \infty)$

When you use it

Use this table to find the domain and range of power functions with positive rational exponents.

Watch out

When the denominator n is even, negative inputs are not allowed in the real numbers.

Drafted from Grade 11 Mathematics, pages 1-82, then checked twice before it went up

Practice questions on unit 1

10 multiple-choice questions written against this chapter, each with its answer and the reason. Try one before you open the answer.

Question 1

Find the domain of the relation R = {(45, 65), (48, 68), (41, 62), (46, 66), (50, 70)}.

A.{41, 45, 46, 48, 50}
B.{62, 65, 66, 68, 70}
C.{41, 45, 46, 48, 50, 62, 65, 66, 68, 70}
D.{45, 48, 41, 46, 50, 65}

Show the answer and why

Answer: A {41, 45, 46, 48, 50}

Why

The domain is the set of all first components of the ordered pairs. The first components are 41, 45, 46, 48 and 50. The set {62, 65, 66, 68, 70} is the range, not the domain.

Question 2

Let R = {(1, 3), (2, 5), (3, 7), (4, 11), (5, 23)}. Which set is the inverse relation R^-1?

A.{(3, 1), (5, 2), (7, 3), (11, 4), (23, 5)}
B.{(1, 3), (2, 5), (3, 7), (4, 11), (5, 23)}
C.{(1, 2), (3, 4), (5, 6), (7, 8), (9, 10)}
D.{(-1, -3), (-2, -5), (-3, -7), (-4, -11), (-5, -23)}

Show the answer and why

Answer: A {(3, 1), (5, 2), (7, 3), (11, 4), (23, 5)}

Why

To form the inverse, swap the order of the two components in each ordered pair. For example, (1, 3) becomes (3, 1). Doing this for all five pairs gives option A.

Question 3

Which of the following is not a power function?

A.f(x) = 3x2
B.g(x) = 5x3
C.n(x) = -4x-3
D.h(x) = 2x

Show the answer and why

Answer: D h(x) = 2x

Why

A power function has the form f(x) = ax^r, where a is a nonzero real number and r is a real number. In h(x) = 2^x, the variable is in the exponent, so it is an exponential function, not a power function. The other three have the form a times x to a power.

Question 4

Using the vertical line test, a vertical line cuts the graph of a relation at two different points. What does this tell you?

A.The relation is a function.
B.The relation is not a function.
C.The relation has no domain.
D.The relation is its own inverse.

Show the answer and why

Answer: B The relation is not a function.

Why

A set of points represents a function if and only if no two points are on the same vertical line. If a vertical line touches the graph twice, one first coordinate has two second coordinates, so the relation is not a function.

Question 5

Abebe buys teff at a shop in Addis Ababa. The relation R = {(2, 80), (3, 120), (2, 95), (4, 160)} lists pairs of (kilograms of teff, birr paid). Is R a function?

A.Yes, because every first component appears only once.
B.Yes, because the second components are all different.
C.No, because 2 is paired with both 80 and 95.
D.No, because 4 is paired with 160.

Show the answer and why

Answer: C No, because 2 is paired with both 80 and 95.

Why

A relation is a function when each first component is related to exactly one second component. Here the first component 2 is paired with two different values, 80 and 95. So R is not a function. The pair (4, 160) does not cause the problem.

Question 6

Let R = {(x, y) : y = x² and x is in {-2, -1, 0, 1, 2}}. Find the range of R.

A.{-2, -1, 0, 1, 2}
B.{0, 1, 4}
C.{1, 4}
D.{-4, -1, 0, 1, 4}

Show the answer and why

Answer: B {0, 1, 4}

Why

Substitute each x value. (-2, 4), (-1, 1), (0, 0), (1, 1) and (2, 4) are the elements of R. The second coordinates are 4, 1, 0, 1 and 4. Written as a set without repeats, the range is {0, 1, 4}.

Question 7

What is the range of the power function f(x) = x⁴?

A.R, all real numbers
B.(0, ∞)
C.[0, ∞)
D.[0, 4]

Show the answer and why

Answer: C [0, ∞)

Why

For an even natural number n, the function xⁿ has domain R and range [0, ∞). Since 4 is even, f(x) = x⁴ has range [0, ∞). The range includes 0, because x = 0 gives 0.

Question 8

A square is cut out of a cardboard. Each side has length x. Which function gives the area of the square in terms of x?

A.A(x) = 4x
B.A(x) = x2
C.A(x) = x3
D.A(x) = 2x2

Show the answer and why

Answer: B A(x) = x2

Why

The area of a square is side times side, so the area is x times x, which is x². This is a power function with a = 1 and r = 2. The value 4x would be the perimeter, not the area.

Question 9

Let R = {(x, y) : x, y ∈ R and y = 3x - 5}. Find the inverse relation R^-1, written as a formula for y in terms of x.

A.y = (x + 5)/3
B.y = 3x + 5
C.y = (x - 5)/3
D.y = x/3 - 5

Show the answer and why

Answer: A y = (x + 5)/3

Why

To find the inverse, swap x and y to get x = 3y - 5. Then solve for y: add 5 to both sides to get x + 5 = 3y, and divide by 3 to get y = (x + 5)/3. Option C is wrong because the 5 must be added, not subtracted.

Question 10

Let R = {(-3, 2), (-2, 3), (-1, 4), (0, 5), (1, 6), (2, 7), (3, 8)}. Find the domain and the range of R^-1.

A.Domain {2, 3, 4, 5, 6, 7, 8}; range {-3, -2, -1, 0, 1, 2, 3}
B.Domain {-3, -2, -1, 0, 1, 2, 3}; range {2, 3, 4, 5, 6, 7, 8}
C.Domain {-3, -2, -1, 0, 1, 2, 3}; range {-3, -2, -1, 0, 1, 2, 3}
D.Domain {2, 3, 4, 5, 6, 7, 8}; range {2, 3, 4, 5, 6, 7, 8}

Show the answer and why

Answer: A Domain {2, 3, 4, 5, 6, 7, 8}; range {-3, -2, -1, 0, 1, 2, 3}

Why

The domain of the inverse is the range of R, and the range of the inverse is the domain of R. The range of R is {2, 3, 4, 5, 6, 7, 8}, and the domain of R is {-3, -2, -1, 0, 1, 2, 3}. So the domain of R^-1 is {2, 3, 4, 5, 6, 7, 8} and its range is {-3, -2, -1, 0, 1, 2, 3}.

The app holds 12 questions on this unit, marked as you go. These are the first 10.