**True or False? Justify your answer with a proof or a counterexample.**

1) A function is always one-to-one.

2) \(f∘g=g∘f\), assuming \(f\) and \(g\) are functions.

**Answer**
- False

3) A relation that passes the horizontal and vertical line tests is a one-to-one function.

4) A relation passing the horizontal line test is a function.

**Answer**
- False

**State the domain and range of the given functions:**

\(f=x^2+2x−3\), \(g=\ln(x−5)\), \(h=\dfrac{1}{x+4}\)

5) h

6) g

**Answer**
- Domain: \(x>5\), Range: all real numbers

7) \(h∘f\)

8) \(g∘f\)

**Answer**
- Domain: \(x>2\) and \(x<−4\), Range: all real numbers

**Find the degree, \(y\)-intercept, and zeros for the following polynomial functions.**

9) \(f(x)=2x^2+9x−5\)

10) \(f(x)=x^3+2x^2−2x\)

**Answer**
- Degree of 3, \(y\)-intercept: \((0,0),\) Zeros: \(0, \,\sqrt{3}−1,\, −1−\sqrt{3}\)

**Simplify the following trigonometric expressions.**

11) \(\dfrac{\tan^2x}{\sec^2x}+{\cos^2x}\)

12) \(\cos^2x-\sin^2x\)

**Answer**
- \(\cos(2x)\)

**Solve the following trigonometric equations on the interval \(θ=[−2π,2π]\) exactly.**

13) \(6\cos 2x−3=0\)

14) \(\sec^2x−2\sec x+1=0\)

**Answer**
- \(0,±2π\)

**Solve the following logarithmic equations.**

15) \(5^x=16\)

16) \(\log_2(x+4)=3\)

**Answer**
- \(4\)

**Are the following functions one-to-one over their domain of existence? Does the function have an inverse? If so, find the inverse \(f^{−1}(x)\) of the function. Justify your answer.**

17) \(f(x)=x^2+2x+1\)

18) \(f(x)=\dfrac{1}{x}\)

**Answer**
- One-to-one; yes, the function has an inverse; inverse: \(f^{−1}(x)=\dfrac{1}{x}\)

**For the following problems, determine the largest domain on which the function is one-to-one and find the inverse on that domain.**

19) \(f(x)=\sqrt{9−x}\)

20) \(f(x)=x^2+3x+4\)

**Answer**
- \(x≥−\frac{3}{2},\quad f^{−1}(x)=−\frac{3}{2}+\frac{1}{2}\sqrt{4x−7}\)

21) A car is racing along a circular track with diameter of 1 mi. A trainer standing in the center of the circle marks his progress every 5 sec. After 5 sec, the trainer has to turn 55° to keep up with the car. How fast is the car traveling?

**For the following problems, consider a restaurant owner who wants to sell T-shirts advertising his brand. He recalls that there is a fixed cost and variable cost, although he does not remember the values. He does know that the T-shirt printing company charges $440 for 20 shirts and $1000 for 100 shirts.**

22) a. Find the equation \(C=f(x)\) that describes the total cost as a function of number of shirts and

b. determine how many shirts he must sell to break even if he sells the shirts for $10 each.

**Answer**
- a. \(C(x)=300+7x\)

b. \(100\) shirts

23) a. Find the inverse function \(x=f^{−1}(C)\) and describe the meaning of this function.

b. Determine how many shirts the owner can buy if he has $8000 to spend.

**For the following problems, consider the population of Ocean City, New Jersey, which is cyclical by season.**

24) The population can be modeled by \(P(t)=82.5−67.5\cos[(π/6)t]\), where \(t\) is time in months (\(t=0\) represents January 1) and \(P\) is population (in thousands). During a year, in what intervals is the population less than 20,000? During what intervals is the population more than 140,000?

**Answer**
- The population is less than 20,000 from December 8 through January 23 and more than 140,000 from May 29 through August 2

25) In reality, the overall population is most likely increasing or decreasing throughout each year. Let’s reformulate the model as \(P(t)=82.5−67.5\cos[(π/6)t]+t\), where t is time in months (\(t=0\) represents January 1) and \(P\) is population (in thousands). When is the first time the population reaches 200,000?

**For the following problems, consider radioactive dating. A human skeleton is found in an archeological dig. Carbon dating is implemented to determine how old the skeleton is by using the equation \(y=e^{rt}\), where \(y\) is the percentage of radiocarbon still present in the material, \(t\) is the number of years passed, and \(r=−0.0001210\) is the decay rate of radiocarbon.**

26) If the skeleton is expected to be 2000 years old, what percentage of radiocarbon should be present?

**Answer**
- 78.51%

27) Find the inverse of the carbon-dating equation. What does it mean? If there is 25% radiocarbon, how old is the skeleton?

## Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.