Math 80 Summer 2015 - Practice Test #4 (Disclaimer: This is not exhaustive of potential test material)
𝑟 !"
Formulas: 𝐴 = 𝑃 1 +
𝐴 = 𝑃𝑒 !" 𝐴 = 𝐴! 2!!/! 𝑃 = 𝑃! 𝑒 !"
𝑘
1. Solve the nonlinear inequality. Graph the solution set on the number line and provide the interval
solution. Use the number line test method.
2𝑥 ! + 7𝑥 − 4
≤0
𝑥+3
2. Graph the conic sections in the plane. Convert to standard form if needed. Identify and label any of the
appropriate features possibly including: any vertices, axis of symmetry, and (if applicable) center.
𝑎) 16𝑥 ! + 25𝑦 ! = 400 𝑏) 𝑥 ! + 𝑦 ! − 8𝑥 − 40𝑦 + 127 = 0
𝑐) 𝑥 = −𝑦 ! + 6𝑦 − 8 𝑑) 4(𝑦 − 2)! − 9(𝑥 + 5)! = 144
3. The half-life of a radioactive substance is known to be 3 years. How long until a 400 mg sample decays
to 10 mg? (Give an exact solution or a decimal rounded to the nearest hundredth of a year).
4. Write as a single logarithm. Simplify.
1
2 log 𝑥 + 6 − log 𝑥 ! − 36 − log 𝑧
2
5. Write as the sum / difference of logarithms. Leave no exponents. Simplify.
𝑒!𝑥
𝑎) log ! 81 𝑥 𝑏) ln !
𝑦
!
!
6. Let 𝑓 𝑥 = 𝑥 + 2𝑥 , 𝑔 𝑥 = 3𝑥 − 1 , ℎ 𝑥 = 𝑥 + 4 . Find the following:
𝑎) 𝑔 ∘ 𝑓 𝑥 𝑏) 𝑓 ∘ 𝑔 −3 𝑐) 𝑔
𝑓
𝑥 and state the domain 𝑑) ℎ!! (𝑥)
7. Solve the following equations.
𝑎) 4 + 3 log 2𝑥 = 16
𝑏) log ! 𝑥 + 2 + log ! 𝑥 + 6 = 5
𝑐) 5!!! = 7
𝑑) ln 3𝑥 − ln 𝑥 − 4 = ln 4
8. Graph the functions. Label any intercepts and asymptotes. State the domain and range of each.
𝑎) 𝑓 𝑥 = log ! 𝑥 + 3 𝑏) 𝑔 𝑥 = −2! + 2
!
9. A population of termites doubles in size every 4 months. Assuming exponential growth, how long until a
given termite population reaches 10 times its original size?
10. Is the given relation a function? Is it one-to-one? Why or why not? { −3 , 4 , 5 , 6 , 1 , 2 , 3, 4 }