MHF 4U Advanced Functions Grade 12
RATIONAL FUNCTIONS
TEST 16
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Multiple Choice
Identify the choice that best completes the statement or answers the
question.
1. State the vertical asymptotes of 𝑓(𝑥 ) =
a. x = 6, x = – 2
b. x = 6, x = 2
1
.
𝑥 2 − 4𝑥−12
c. x = – 6, x = – 2
d. x = – 6, x = 2
2. What is the domain of the reciprocal function of f(x) = 5x + 1?
1
1
a. {𝑥 ∈ ℝ| 𝑥 ≥ − }
c. {𝑥 ∈ ℝ| 𝑥 ≠ − }
b. {𝑥 ∈ ℝ| 𝑥 ≤ − }
d. {𝑥 ∈ ℝ}
5
1
5
5
3. Identify any holes in the graph of 𝑓(𝑥 ) =
a. 𝑥 =
1
7
b. 𝑥 = −
1
7
c. 𝑥 = 7
𝑥+7
d. 𝑥 = −7
4. What is the horizontal asymptote of 𝑓 (𝑥 ) =
a. 𝑦 =
2
5
b. 𝑦 =
7
2
c. 𝑦 = −
.
7𝑥 + 49
7
3
2𝑥− 7
5𝑥 + 3
d. 𝑦 = −
3
5
5. Solve
a. 𝑥 =
4𝑥
𝑥− 2
1
3𝑥 − 2
𝑥−2
for x.
b. 𝑥 = − 2, 2
2
6. Solve
=
𝑥− 6
𝑥+ 2
=
a. 𝑥 = −2, −1
𝑥− 3
𝑥+ 1
c. 𝑥 = −2
d. 𝑥 = 2
for x.
b. 𝑥 = 1
c. 𝑥 = −1
d. 𝑥 = 0
7. Which inequality is equivalent to
𝑥+ 2
𝑥−3
a.
<
2𝑥 2 + 4𝑥 − 10
(𝑥 − 3)(𝑥 + 1)
b. (𝑥
2𝑥 − 10
− 3)(𝑥 + 1)
𝑥+ 4
?
𝑥+ 1
<0
c. (𝑥
<0
d. (𝑥
𝑥+7
− 3)(𝑥 + 1)
2𝑥+ 7
− 3)(𝑥 + 1)
8. Find all values that make the inequality
a. −2 ≤ 𝑥 < 2, 3 ≤ 𝑥 < 6
b. 𝑥 < −2, 2 ≤ 𝑥 ≤ 3, 𝑥 > 6
<0
<0
−𝑥
𝑥− 6
≤
1
𝑥− 2
true.
c. 𝑥 ≤ −2, 2 < 𝑥 ≤ 3, 𝑥 ≥ 6
d. 𝑥 ≤ −2, 2 < 𝑥 ≤ 3, 𝑥 > 6
9. What is the average rate of change of 𝑓 (𝑥 ) =
𝑥−5
𝑥+3
from x = – 2 to x =
4?
a.
24
b.
7
8
7
c. −
25
21
d. −
11
3
10. Which of these functions has a hole in the graph?
a. 𝑓(𝑥 ) =
b. 𝑓(𝑥 ) =
𝑥− 3
𝑥+ 4
𝑥 2 − 3𝑥 + 2
𝑥+ 2
c. 𝑓 (𝑥 ) =
d. 𝑓(𝑥 ) =
𝑥2 − 9
𝑥+ 1
𝑥 2 − 5𝑥 − 6
𝑥+1
Problem Solving
11. Sketch the graph of 𝑓(𝑥 ) = 𝑥 2 − 2𝑥 and the graph of the reciprocal
function. Be sure to include equations of asymptotes and coordinates of
intercepts. /5 APP
12. Write a formula or describe a method for identifying each of these
characteristics of 𝑓(𝑥 ) =
𝑎𝑥 + 𝑏
𝑐𝑥 + 𝑑
.
/7 COMM
a. x-intercepts
b. y-intercept
c. vertical asymptotes
d. horizontal asymptotes
e. holes in the graph
f. intervals where f(x) is positive or negative
g. Describe f(x) if the horizontal asymptote is y = 0.
13. A passenger jet flew from Dallas, Texas, to Germany and back. The
return flight took 4 hours less than it took to go to Germany. The
difference in flight time caused by winds over the ocean on the return
trip averaged 50 km/h. Assume there was no wind factor on the trip to
Germany. The round trip is 9900 km.
a) What was the average rate of plane in still air?
b) How long did the round trip take?
/1 APP
/4 APP
14. Solve: 2𝑥 − 1 <
𝑥+ 7
𝑥+ 1
/4 TH
15. List the transformations necessary to transform 𝑦 =
3
2𝑥 − 1
− 2.
/2 TH
𝑥+ 4
2− 𝑥
to 𝑦 =