# 1.5 Exponential and logarithmic functions  (Page 10/17)

 Page 10 / 17

h

g

Domain: $x>5,$ range: all real numbers

$h\circ f$

$g\circ f$

Domain: $x>2$ and $x<-4,$ range: all real numbers

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

$f\left(x\right)=2{x}^{2}+9x-5$

$f\left(x\right)={x}^{3}+2{x}^{2}-2x$

Degree of 3, $y$ -intercept: 0, zeros: 0, $\sqrt{3}-1,-1-\sqrt{3}$

Simplify the following trigonometric expressions.

$\frac{{\text{tan}}^{2}x}{{\text{sec}}^{2}x}+{\text{cos}}^{2}x$

$\text{cos}\left(2x\right)={\text{sin}}^{2}x$

$\text{cos}\left(2x\right)$ or $\frac{1}{2}\left(\text{cos}\left(2x\right)+1\right)$

Solve the following trigonometric equations on the interval $\theta =\left[-2\pi ,2\pi \right]$ exactly.

$6{\text{cos}}^{2}x-3=0$

${\text{sec}}^{2}x-2\phantom{\rule{0.1em}{0ex}}\text{sec}\phantom{\rule{0.1em}{0ex}}x+1=0$

$0,±2\pi$

Solve the following logarithmic equations.

${5}^{x}=16$

${\text{log}}_{2}\left(x+4\right)=3$

4

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

$f\left(x\right)={x}^{2}+2x+1$

$f\left(x\right)=\frac{1}{x}$

One-to-one; yes, the function has an inverse; inverse: ${f}^{-1}\left(x\right)=\frac{1}{y}$

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

$f\left(x\right)=\sqrt{9-x}$

$f\left(x\right)={x}^{2}+3x+4$

$x\ge -\frac{3}{2},{f}^{-1}\left(x\right)=-\frac{3}{2}+\frac{1}{2}\sqrt{4y-7}$

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.

a. Find the equation $C=f\left(x\right)$ 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. a. $C\left(x\right)=300+7x$ b. 100 shirts a. Find the inverse function $x={f}^{-1}\left(C\right)$ 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.

The population can be modeled by $P\left(t\right)=82.5-67.5\phantom{\rule{0.1em}{0ex}}\text{cos}\left[\left(\pi \text{/}6\right)t\right],$ where $t$ is time in months $\left(t=0$ 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?

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

In reality, the overall population is most likely increasing or decreasing throughout each year. Let’s reformulate the model as $P\left(t\right)=82.5-67.5\phantom{\rule{0.1em}{0ex}}\text{cos}\left[\left(\pi \text{/}6\right)t\right]+t,$ 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.

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

78.51%

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

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