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MCR 3U Grade 11 University
FUNCTIONS AND TRANSFORMATIONS
Review
Relation: a rule that shows how two quantities are related.
Mathematically, a relation is a rule that associates each element x from
one set, called domain, with one or more elements y from the other set,
called range, to create a set of ordered pairs (x, y). Thus, a relation is a
set of ordered pairs.
A relation that assigns to each element in the domain exactly one
element in the range is called a function. We write: 𝑦 = 𝑓 (𝑥 ) and read
as: function f at value x is equal to y.
We can think of a function as a machine.
Domain: the set of the first elements in a relation. Range: the set of the
second elements in a relation.
1. For each relation, determine the domain, range, and whether the
relation is a function. Explain your reasoning.
a) {(−3, 0), (−1, 1), (0, 1), (4, 5), (0, 6)}
b)
c)
d)
e)
f)
g)
h)
i) 𝑦 = 4 − 𝑥
j) 𝑦 = −(𝑥 − 3)2 + 5
k) 𝑦 = √𝑥 − 4
l) 𝑥 2 + 𝑦 2 = 16
2. Sketch the graph of a function whose domain is the set of all real
numbers and whose range is the set of all real numbers less than or equal
to 3.
3. a) Graph the function 𝑓(𝑥 ) = −2(𝑥 + 1)2 + 3.
b) Determine 𝑓 (2 − 𝑥 ).
4. A teacher asked her students to think of a number, multiply it by 5,
and subtract the product from 20. Then she asked them to multiply the
resulting difference by the number they first thought of.
a) Use function notation to express the final answer in terms of the
original number.
b) Determine the outputs for the input numbers 1, – 1, and 7.
c) Determine the maximum result possible.
5. If 𝑓(𝑥 ) = 𝑥 2 + 3𝑥 − 5 and 𝑔(𝑥 ) = 2𝑥 − 3, determine each
a) 𝑓(1 − 2𝑎)
1
b) 𝑓 (𝑔 (− 𝑥 + 2))
2
c) x when 𝑓(𝑥 ) = 𝑔(𝑥 )
6. Graph each function and state its domain and range.
1
a) 𝑓(𝑥 ) =
𝑥
b) 𝑓 (𝑥 ) = √𝑥
c) 𝑓(𝑥 ) = |𝑥|
7. Determine the domain and range for each.
a) A parabola has a vertex at (−2, 5), and 𝑦 = 5 is its maximum value.
b) A circle has a center at (−2, 5) and a radius of 4.
8. A farmer has 600 m of fencing to enclose a rectangular area and
divide it into three sections as shown.
a) Write an equation to express the total area enclosed as a function of
the width.
b) Determine the domain and range of this area function.
c) Determine the dimensions that give the maximum area.
9. A ball is thrown upward from the roof of a building 60 m tall. The
ball reaches a height of 80 m above the ground after 2 s and hits the
ground 6 s after being thrown.
a) Sketch a graph that shows the height of the ball as a function of time.
b) State the domain and range of the function.
c) Determine an equation for the function.
10. State the domain and range of each function.
a) 𝑓(𝑥 ) = 2(𝑥 − 1)2 + 3
b) 𝑓 (𝑥 ) = √2𝑥 + 4
11.
a) Sketch 𝑓(𝑥 ) = 𝑥 2 − 4𝑥.
b) Write equations for 𝑔(𝑥 ) = − 𝑓(𝑥 ) and ℎ(𝑥 ) = 𝑓 (−𝑥 ).
c) State the domain and range of 𝑔(𝑥 ) and ℎ(𝑥 ).
d) Sketch the graphs of 𝑦 = 𝑔(𝑥 ) and 𝑦 = ℎ(𝑥 ). Describe the
relationship between each graph and the graph of 𝑦 = 𝑓(𝑥 ).
12. Three transformations are applied to 𝑦 = 𝑥 2 : a vertical stretch by
the factor of 2, a translation 3 units right, and a translation 4 units down.
Is the order of transformations important?
13. The point (1, 4) is on the graph of 𝑦 = 𝑓 (𝑥 ). Determine the
coordinates of the image of this point on the graph of 𝑦 = 3𝑓(−4𝑥 +
4) − 2.
1
14. Given 𝑓(𝑥 ) = 𝑥 2 , graph the function 𝑔(𝑥 ) = −2𝑓 ( 𝑥 + 1) using
3
mapping notation, then state the domain, range and invariant points.
State the equation of 𝑔(𝑥 ).
1
1
3
15. Given 𝑓(𝑥 ) = , graph the function 𝑔(𝑥 ) = −𝑓 (− 𝑥 + ) − 2
𝑥
2
2
using mapping notation, then state the domain, range and invariant
points. State the equation of 𝑔(𝑥 ).
16. In each case, write the equation for the transformed function, sketch
its graph, and state its domain and range.
1
a) The graph of 𝑓 (𝑥 ) = √𝑥 is compressed horizontally by a factor ,
2
reflected in the y-axis, and translated 3 units right and 2 units down.
1
b) The graph of 𝑓 (𝑥 ) = is stretched vertically by a factor 3 reflected in
𝑥
the x-axis, and translated 4 units left and 1 units up.
17. If 𝑓(𝑥 ) = (𝑥 − 4)(𝑥 + 3), determine the x-intercepts of each
function.
a) 𝑦 = 𝑓 (𝑥 )
b) 𝑦 = −2𝑓(𝑥 )
1
c) 𝑦 = 𝑓 (− 𝑥)
2
d) 𝑦 = 𝑓(−𝑥 − 1)
18. A function 𝑓 (𝑥 ) has domain {𝑥 ∊ ℝ| 𝑥 ≥ −4} and range {𝑦 ∊
ℝ| 𝑦 < −1}. Determine the domain and range of each function.
a) 𝑦 = 2𝑓 (𝑥 )
b) 𝑦 = 𝑓(−𝑥 )
c) 𝑦 = 3𝑓 (𝑥 ) + 4
d) 𝑦 = −2𝑓(−𝑥 + 5) + 1
19.
a) Determine the equation that represents the inverse of the
3
function 𝑓(𝑥 ) = 𝑥 + 2.
2
b) Graph the original function and its inverse.
c) State the domain and range for each relation. What do you notice
about the domain and range of 𝑓(𝑥 ) and 𝑓 −1 (𝑥 ).
d) Find 𝑓 −1 (𝑓(𝑥 )) and 𝑓(𝑓 −1 (𝑥 ))
20.
a) Determine the equation that represents the inverse of the
function 𝑓(𝑥 ) = 𝑥 2 + 1
b) Graph the original function and its inverse.
c) State the domain and range for each relation. What do you notice
about the domain and range of 𝑓(𝑥 ) and 𝑓 −1 (𝑥 ).