Unit notes and practice questions

Grade 11 Mathematics unit 2: Rational Expressions and Rational Functions

Everything here is for unit 2 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.

4
Cards: 10
Practice questions on unit 2: 83–130
The textbook pages for unit 2

What unit 2 covers

1.Rational Expressions
2.Rational Equations and Rational Inequalities
3.Rational Functions and Their Graphs
4.Applications

Short notes for unit 2

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

Asymptotes of Rational FunctionsReference tabledeg(p) < deg(q) · deg(p) = deg(q) · deg(p) = deg(q) + 1

Degree condition	Asymptote type	Equation
$\text{deg}(p) < \text{deg}(q)$	Horizontal asymptote	$y = 0$
$\text{deg}(p) = \text{deg}(q)$	Horizontal asymptote	$y = \frac{a_n}{b_n}$
$\text{deg}(p) = \text{deg}(q) + 1$	Oblique asymptote	$y = mx + b$ via polynomial division

When you use it

Use this to determine horizontal or oblique asymptotes when graphing rational functions.

Watch out

Oblique asymptotes occur only when the numerator degree is exactly one greater than the denominator degree.

Drafted from Grade 11 Mathematics, pages 83-130, then checked twice before it went up

Operations on Rational ExpressionsReference tableAddition · Subtraction · Multiplication · +1

Operation	Rule	Condition
Addition	$\frac{p(x)}{q(x)} + \frac{r(x)}{s(x)} = \frac{p(x)s(x) + r(x)q(x)}{q(x)s(x)}$	$q(x)$ and $s(x)$ are not zero
Subtraction	$\frac{p(x)}{q(x)} - \frac{r(x)}{s(x)} = \frac{p(x)s(x) - r(x)q(x)}{q(x)s(x)}$	$q(x)$ and $s(x)$ are not zero
Multiplication	$\frac{p(x)}{q(x)} \cdot \frac{r(x)}{s(x)} = \frac{p(x)r(x)}{q(x)s(x)}$	$q(x)$ and $s(x)$ are not zero
Division	$\frac{p(x)}{q(x)} \div \frac{r(x)}{s(x)} = \frac{p(x)s(x)}{q(x)r(x)}$	$q(x)$ , $s(x)$ , and $r(x)$ are not zero

When you use it

Use these rules to combine and simplify sums, differences, products, and quotients of rational expressions.

Watch out

In division, the numerator of the divisor fraction also cannot be zero because it moves to the denominator.

Drafted from Grade 11 Mathematics, pages 83-130, then checked twice before it went up

Rational Expression DefinitionRule

When you use it

Use this to identify whether an algebraic expression is a rational expression.

Watch out

Both the numerator and the denominator must be polynomials, and the denominator cannot be the zero polynomial.

Drafted from Grade 11 Mathematics, pages 83-130, then checked twice before it went up

Work FormulaFormula

W = r t

W = r t

$W$: Work completed $1$
$r$: Combined work rate $\text{h}^{-1}$
$t$: Time taken $\text{h}$

When you use it

Use this when multiple workers or pipes complete a task together at constant rates.

Watch out

Add individual rates of work per unit time, never add completion times directly.

Drafted from Grade 11 Mathematics, pages 83-130, then checked twice before it went up

Practice questions on unit 2

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

Question 1

Across which line is the graph of a function reflected to obtain the graph of its inverse function?

A.The line y = x
B.The line y = -x
C.The line y = 0 (x-axis)
D.The line x = 0 (y-axis)

Show the answer and why

Answer: A The line y = x

Why

The graph of the inverse of a given function is obtained by reflecting the graph of the function with respect to the line y = x.

Question 2

By definition, an expression written in the form p(x) / q(x) is a rational expression if p(x) and q(x) are what type of expressions, with q(x) ≠ 0?

A.Polynomial expressions
B.Absolute value expressions
C.Radical expressions
D.Logarithmic expressions

Show the answer and why

Answer: A Polynomial expressions

Why

A rational expression is defined as a quotient of two polynomials, p(x) / q(x), where q(x) ≠ 0.

Question 3

If R is a relation, which of the following is always true about its inverse relation R^-1?

A.Domain of R-1 = Range of R
B.Domain of R-1 = Domain of R
C.Range of R-1 = Range of R
D.Domain of R-1 = Range of R-1

Show the answer and why

Answer: A Domain of R-1 = Range of R

Why

For any relation R, the domain of R^-1 equals the range of R, and the range of R^-1 equals the domain of R.

Question 4

What is the value of the signum function sgn(x) when x > 0?

A.1
B.0
C.-1
D.x

Show the answer and why

Answer: A 1

Why

By definition, sgn(x) = 1 if x > 0, 0 if x = 0, and -1 if x < 0.

Question 5

Given the linear function f(x) = 3x - 1, what is its inverse function f^-1(x)?

A.(1/3)x + (1/3)
B.(1/3)x - (1/3)
C.-3x + 1
D.3x + 1

Show the answer and why

Answer: A (1/3)x + (1/3)

Why

Let y = 3x - 1. Interchanging variables gives x = 3y - 1. Solving for y gives 3y = x + 1, so y = (1/3)x + (1/3).

Question 6

A car rental company in Addis Ababa charges a flat fee of 300 Birr plus 15 Birr per kilometer to rent a van. What is the total cost for an 80 kilometer trip?

A.1500 Birr
B.1200 Birr
C.1800 Birr
D.1450 Birr

Show the answer and why

Answer: A 1500 Birr

Why

The cost function is y = 15x + 300. For x = 80, y = (15 × 80) + 300 = 1200 + 300 = 1500 Birr.

Question 7

The force applied to a spring varies directly with the distance it stretches. If applying 30 N of force stretches the spring 3 m, how far will it stretch when 5 N of force is applied?

A.0.5 m
B.1.5 m
C.2.0 m
D.0.2 m

Show the answer and why

Answer: A 0.5 m

Why

Using F = kd, we find 30 = k(3), so k = 10 N/m. When F = 5 N, d = F / k = 5 / 10 = 0.5 m.

Question 8

Given the function f(x) = 2 - 3 sgn(x), what is the value of f(5)?

A.-1
B.2
C.5
D.-3

Show the answer and why

Answer: A -1

Why

Since 5 > 0, sgn(5) = 1. Therefore, f(5) = 2 - 3(1) = 2 - 3 = -1.

Question 9

A vendor buys 300 newspapers at 6 Birr each. He sells 200 newspapers at 10 Birr each, and sells 50 unsold newspapers to a wastepaper dealer at 4 Birr each. What is his total profit for that day?

A.400 Birr
B.200 Birr
C.600 Birr
D.800 Birr

Show the answer and why

Answer: A 400 Birr

Why

Total cost = 300 × 6 = 1800 Birr. Total revenue = (200 × 10) + (50 × 4) = 2000 + 200 = 2200 Birr. Profit = 2200 - 1800 = 400 Birr.

Question 10

In a closed economy model where E = C + I + G and equilibrium occurs at E = Y, consumption is C = 15 + 0.90Y, investment is I = 20 + 0.05Y, and government spending is G = 25. What is the equilibrium income Y?