1. Functions and Graphs

# Chapter 1 Review Exercises

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

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

**2. **, assuming and are functions.

#### Solution

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.

#### Solution

False

For the following problems, state the domain and range of the given functions:

**5. **

**6. **

#### Solution

Domain: , Range: all real numbers

**7. **

**8. **

#### Solution

Domain: or , Range: all real numbers

Find the degree, -intercept, and zeros for the following polynomial functions.

**9. **

**10. **

#### Solution

Degree of 3, -intercept: 0, Zeros: 0,

Simplify the following trigonometric expressions.

**11. **

**12. **

#### Solution

or

Solve the following trigonometric equations on the interval exactly.

**13. **

**14. **

#### Solution

Solve the following logarithmic equations.

**15. **

**16. **

#### Solution

4

Are the following functions one-to-one over their domain of existence? Does the function have an inverse? If so, find the inverse of the function. Justify your answer.

**17. **

**18. **

#### Solution

One-to-one; yes, the function has an inverse; inverse:

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

**19. **

**20. **

#### Solution

**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 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 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.

#### Solution

a. b. 100 shirts

**23. **a. Find the inverse function 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 , where is time in months ( represents January 1) and 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?

#### Solution

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 , where is time in months ( represents January 1) and 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 , where is the percentage of radiocarbon still present in the material, is the number of years passed, and is the decay rate of radiocarbon.

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

#### Solution

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?