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1.R: Chapter 1 Review Exercises

Last updated

Apr 22, 2024
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- 1.5E: Exercises for Section 1.5
- 2: Limits

Gilbert Strang & Edwin “Jed” Herman
OpenStax

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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.

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