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Math 107 Quiz 5
1. (4 pts) Which of these graphs represent a one-to-one function?
A one-to-one function is a function where each input value for 𝑦 has a single,
unique output value for 𝑥.
If you draw a straight horizontal line across the graph of any function, they will be
one-to-one if the line only intersects at a single point.
By this reasoning, only the graphs represented in option (A) and option (C) are a
match.
Therefore, the answers are (A) and (C).
2. (6 pts) Based on data about the growth of a variety of ornamental cherry trees,
the following logarithmic model about these trees was determined:
ℎ(𝑡) = 6.108 𝑙𝑛(𝑡) + 6.099, where 𝑡 = age of tree in years and ℎ(𝑡) = height of
tree, in feet.
Using this model,
(a)
At age 3 years, how tall is this type of ornamental cherry tree, to the
nearest tenth of a foot?
We have to insert 𝑡 = 3 in the above logarithmic model:
⇒ ℎ(3) = 6.108 ln(3) + 6.099 = 12.809 ≈ 12.8
Therefore, the answer is 12.8 feet, approximated to the nearest tenth of a
foot.
(b)
At age 12 years, how tall is this type of ornamental cherry tree, to the
nearest tenth of a foot?
This sub-question is almost identical to part (a), except for the value of
age.
So, we insert 𝑡 = 12 into the model and evaluate the height:
⇒ ℎ(12) = 6.108 ln(12) + 6.099 = 21.2768 ≈ 21.3
Therefore, the answer is 21.3 feet, approximated to the nearest tenth of a
foot.
3. (4 pts) Convert to a logarithmic equation: 5x = 645.
To convert this exponential equation into a logarithmic one, we can first apply
log 5 () to both sides of the equation:
⇒ log 5 (5𝑥 ) = log 5(645)
The logarithm rule that we will first use is the power rule log 𝑏 (𝑥 𝑦 ) = 𝑦 ⋅ log 𝑏 (𝑥):
⇒ 𝑥 log 5 (5) = log 5 (645)
Then, we use the logarithm of base rule log 𝑏 (𝑏) = 1:
⇒ 𝑥 = log 5 (645)
∴ log 5 (645) = 𝑥
Therefore, the answer is A. log 5 645 = 𝑥.
4. (8 pts) Solve the equation. Check all proposed solutions. Show work in solving and
in checking, and state your final conclusion.
√𝑥 − 1 = 3 − 𝑥
First, let’s square both sides of the equation:
2
⇒ (√𝑥 − 1) = (3 − 𝑥)2
⇒ 𝑥 − 1 = 9 − 6𝑥 + 𝑥 2
Then, we simplify, and make the left-hand side equal to 0:
⇒ 0 = 𝑥 2 − 7𝑥 + 10
...