Grade by Grade
A. Characteristics of Functions · Functions and Equivalent Expressions

Functions and Equivalent Expressions: Unit Review

A1.1 A1.2 A1.3 A1.4 A1.5 A1.6 A1.7 A1.8 A1.9 A3.1 A3.2 A3.3 A3.4About 40 minutes
Learning goals
  • I can show what I learned about functions, inverses, transformations and equivalent expressions.

Words to know

function
A relation in which each input has exactly one output.
inverse
The relation that undoes a function; its graph is the reflection in y = x.
transformation
A translation, stretch, compression or reflection of a graph.
restriction
A value of the variable that is not allowed, such as one that makes a denominator zero.
equivalent expressions
Expressions that have the same value for every allowed value of the variable.

Let's review!

In this unit you learned exactly what a function is and how to write it with function notation. You described domain and range with set notation and met the four parent functions x, x², √x and 1/x. You found inverses from tables, graphs and equations, and restricted the domain of a quadratic so its inverse is a function. You moved, stretched and flipped graphs with y = af(k(x − d)) + c. Then you built your algebra toolkit: polynomials, rational expressions with restrictions, and tests for equivalent expressions, including equivalent forms of radicals.

This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Do one or two sections at a time. For graphing questions, sketch on grid paper and label key points. For algebra questions, always state restrictions and check your answer by substituting a number (not 0 or 1). For every question you get wrong, write down why (sign error? wrong direction for a shift? forgot a restriction? factoring?) and go back to that lesson's worked examples. Finish with the 10-question quiz.

Functionseach input → exactly one outputvertical line testf(a) = b means (a, b) is on the graphDomain and rangeno ÷ 0; no √ of a negativeparents: x, x², √x, 1/xcontext limits (time ≥ 0, whole numbers)Inversesundo in reverse order; swap x and yreflect in y = xD and R swap; restrict a parabolaTransformationsy = af(k(x − d)) + c(x, y) → (x/k + d, ay + c)stretch/reflect first, then translate
The function ideas from Lessons 1 to 10 on one page.
Polynomialsevery term × every term(a + b)² ≠ a² + b²collect like termsRadicals√(ab) = √a × √b√72 = 6√2add only like radicalsRational expressionsfactor firstrestrictions from ALLdenominators (and divisors)Equivalent?simplify to prove; one counterexample to disprove
The algebra toolkit from Lessons 11 to 15.
Just for fun

By now you can turn y = x² into y = −3(x + 2)² + 4 without breaking a sweat. That parabola has been reflected, stretched and moved twice, and it still has exactly one output for every input. Respect.

Relations and Functions: What Makes a Function?

1

Is {(2, 5), (3, 5), (2, 7)} a function?

2

Does x² + y² = 16 define a function? Explain.

3

Is "each student → their locker number" a function if every student has one locker?

Function Notation: Naming, Evaluating and Reading Functions

1

f(x) = 2x² − 5x + 1. Find f(−2).

2

g(x) = x² − 3. Find g(x + 2) and simplify.

3

h(x) = 4x − 9. Solve h(x) = 15.

Domain and Range: Inputs, Outputs and Real-World Limits

1

State the domain and range of f(x) = −2(x − 1)² + 8.

2

A graph runs from a closed dot at (−4, 2) to an open dot at (3, −5), along a straight line. State the domain and range.

3

A school sells t-shirts for $15; at most 200 are printed. Revenue R(n) = 15n. State a reasonable domain.

The Parent Functions: x, x², √x and 1/x

1

State the domain of f(x) = √(2x − 10).

2

State the domain of g(x) = 1/(x + 7).

3

Which parent function has range {y ∈ ℝ | y ≠ 0}?

4

Name the asymptotes of y = 1/x.

Inverse Functions: Undoing a Process, Tables and Reflections

1

A function divides by 2, then adds 7. Describe its inverse.

2

If (−3, 10) is on f, which point is on the inverse?

3

A function has domain {x ∈ ℝ | x ≥ 1} and range {y ∈ ℝ | y ≤ 4}. State the domain and range of its inverse.

Finding the Equation of an Inverse: Linear Functions

1

Find the inverse of f(x) = 6x − 5.

2

Find the inverse of g(x) = (x + 8)/3.

3

A line has slope 2/3. What is the slope of its inverse?

Inverses of Quadratic Functions and Restricted Domains

1

Find the inverse of f(x) = (x + 3)² − 2.

2

How should f(x) = (x + 3)² − 2 be restricted so that f⁻¹(x) = −3 + √(x + 2)?

3

Is the inverse of y = x² a function? Explain.

Translations: y = f(x − d) + c

1

Describe y = √(x − 6) + 2 compared with y = √x.

2

State the asymptotes of y = 1/(x + 5) − 3.

3

Write y = x² translated 4 left and 7 down.

Stretches, Compressions and Reflections: y = af(kx)

1

Describe y = −3x² compared with y = x².

2

Describe y = √(5x).

3

The point (6, 4) is on y = f(x). Find its image on y = (1/2)f(−2x).

Combining Transformations: Sketching y = af(k(x − d)) + c

1

For y = √(4x + 8), find k and d.

2

Map (4, 2) on y = √x to y = −2√(x − 1) + 5.

3

State the domain and range of y = −2√(x − 1) + 5.

4

Write y = 1/x stretched vertically by 4, then moved 3 right and 1 down.

Adding, Subtracting and Multiplying Polynomials

1

Simplify (3x² − x + 4) − (x² + 5x − 2).

2

Expand (x − 1)(x² + 2x − 3).

3

Expand (x + 2)³.

Rational Expressions: Simplifying and Stating Restrictions

1

Simplify (x² − 7x + 10)/(x² − 25).

2

Simplify (6 − 2x)/(x − 3).

3

State the restrictions on (x + 1)/(x² + 3x).

Multiplying and Dividing Rational Expressions

1

Simplify (x² − 1)/(4x) × 8x²/(x + 1).

2

Simplify (x + 2)/(x − 5) ÷ (x² − 4)/(x² − 25).

3

Which extra restriction does division add in (3/x) ÷ ((x − 6)/(x + 1))?

Adding and Subtracting Rational Expressions

1

Simplify 3/(2x) + 1/(5x).

2

Simplify 4/(x + 1) − 3/(x − 2).

3

Simplify 2/(x² − 9) + 1/(x − 3).

Are They Equivalent? Testing Algebraic Expressions

1

Are (x + 5)² and x² + 25 equivalent?

2

Are 2x(x − 3) + 6x and 2x² equivalent?

3

Are (x² − x)/x and x − 1 equivalent?

4

Are √108 and 6√3 equivalent?

Check your understanding

Quiz

1.

Which relation is NOT a function?

2.

If f(x) = x² − 4x, what is f(−3)?

3.

What is the domain of g(x) = √(x + 6)?

4.

What is the inverse of f(x) = 5x + 2?

5.

The graph of an inverse is a reflection of the function's graph in:

6.

Compared with y = √x, the graph of y = √(x + 4) − 1 is:

7.

What is the range of y = −2√(x − 3) + 6?

8.

Simplify √50 + √18.

9.

Simplify (x² − 16)/(x² + x − 12).

10.

Which pair of expressions is equivalent?

For parents and teachers

Use the review to find what to revisit:

  • Confusing relations and functions, or using a horizontal line instead of a vertical one → Lesson 1 (Relations and Functions).
  • Sign errors when substituting, or mixing up f(x + 1) and f(x) + 1 → Lesson 2 (Function Notation).
  • Domain and range swapped, or open/closed endpoints wrong → Lesson 3 (Domain and Range).
  • Domain of √ and 1/x functions, asymptotes → Lesson 4 (The Parent Functions).
  • Undoing steps in the wrong order, reflecting in the wrong line → Lesson 5 (Inverse Functions).
  • Algebra errors finding a linear inverse → Lesson 6 (Inverses of Linear Functions).
  • Missing the ± or the domain restriction for a quadratic inverse → Lesson 7 (Inverses of Quadratic Functions).
  • Shifting the wrong way horizontally → Lesson 8 (Translations).
  • Multiplying x by k instead of dividing → Lesson 9 (Stretches, Compressions and Reflections).
  • Forgetting to factor k out, or translating before stretching → Lesson 10 (Combining Transformations).
  • Errors in expanding or subtracting polynomials → Lesson 11 (Polynomials).
  • Missing restrictions, cancelling terms → Lesson 12 (Simplifying Rational Expressions).
  • Forgetting the divisor's numerator restriction → Lesson 13 (Multiplying and Dividing).
  • Common denominators and bracket errors when subtracting → Lesson 14 (Adding and Subtracting).
  • Believing a few matching values prove equivalence, or radical forms that are not fully simplified → Lesson 15 (Equivalent Expressions). For more radical practice, use the lesson "Exact Answers: Simplifying and Operating with Radicals" in the Quadratic Functions unit.

Next in MCR3U, the Quadratic Functions unit (expectations A2.1 to A2.5, plus a full radicals lesson for A3.2) uses these skills to find zeros, maximum and minimum values, and intersections of lines and parabolas. In Grade 12, Advanced Functions (MHF4U) extends transformations and inverses to polynomial, rational, exponential, logarithmic and trigonometric functions.