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What is the domain of this exponential function? $\newline$ $y = 5^x$ $\newline$ Choices: $\newline$ (A) all real numbers $\newline$ (B) \{x | x > 0\} $\newline$ (C) $\{x | x \leq 0\}$ $\newline$ (D) $\{x | x \geq 0\}$ $\newline$ (E) \{x | x < 0\}

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Domain and range of exponential functions: equations

Full solution

Q. What is the domain of this exponential function?

\newline

y = 5^x

\newline

Choices:

\newline

(A) all real numbers

\newline

(B)

\{x | x > 0\}

\newline

(C)

\{x | x \leq 0\}

\newline

(D)

\{x | x \geq 0\}

\newline

(E)

\{x | x < 0\}

Understand domain of function: Understand what the domain of a function represents. $\newline$ The domain of a function is the set of all possible input values ( $x$ -values) for which the function is defined.
Determine possible $x$ -values: Determine the possible $x$ -values for $y = 5^x$ . $\newline$ Since $5^x$ is defined for all real numbers (because any real number can be an exponent), the domain includes all real numbers.
Match domain to choices: Match the domain to the given choices. $\newline$ Since the domain includes all real numbers, the correct choice is $(A)$ all real numbers.

More problems from Domain and range of exponential functions: equations

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Each year, Francesca earns a salary that is

2 \%

higher than her previous year's salary. In her first

5

years at this job, she earned a total of

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\newline

What was Francesca's salary in her

1^{\text {st }}

year at this job?

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Round your final answer to the nearest thousand.

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The following formula gives the temperature's measure in degrees Fahrenheit

F

C

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F=\frac{9}{5} C+32

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Rearrange the formula to highlight the measure in degrees Celsius.

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C=

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Question

2010

, the town of Fall River has been experiencing a growth in population.

\newline

The relationship between the elapsed time,

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, in years, since

2010

and the town's population,

P(t)

, is modeled by the following function.

\newline

P(t)=36,800 \cdot 2^{\frac{t}{25}}

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According to the model, what will the population of Fall River be in

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Round your answer, if necessary, to the nearest whole number.

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A huge ice glacier in the Himalayas initially covered an area of

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\newline

The relationship between

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t

, the number of years the glacier has been melting, is modeled by the following equation.

\newline

A=45 e^{-0.05 t}

\newline

How many years will it take for the area of the glacier to decrease to

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Give an exact answer expressed as a natural logarithm.

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Question

Enzo is studying the black bear population at a large national park.

\newline

He finds that the relationship between the elapsed time

t

, in years, since the beginning of the study, and the black bear population

B(t)

, in the park is modeled by the following function.

\newline

B(t)=2500 \cdot 2^{0.01 t}

\newline

According to the model, what will the black bear population be at that national park in

25

years? Round your answer, if necessary, to the nearest whole number.

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Find the following trigonometric values.

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Express your answers exactly.

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Question

Valeria is playing with her accordion.

\newline

The length of the accordion

A(t)

\mathrm{cm}

) after she starts playing as a function of time

t

(in seconds) can be modeled by a sinusoidal expression of the form

a \cdot \cos (b \cdot t)+d

\newline

t=0

, when she starts playing, the accordion is

15 \mathrm{~cm}

long, which is the shortest it gets.

1

5

seconds later the accordion is at its average length of

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\newline

A(t)

\newline

t

should be in radians.

\newline

A(t)=\square

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Question

On the first day of spring, an entire field of flowering trees blossoms. The population of locusts consuming these flowers rapidly increases as the trees blossom. The locust population gains

87 \%

of its size every

2

4

days, and can be modeled by a function,

L

, which depends on the amount of time,

t

\newline

Before the first day of spring, there were

1100

locusts in the population.

\newline

Write a function that models the locust population

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\newline

L(t)=

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Question

Aleksandra started studying how the number of branches on her tree grows over time. Every

1

5

years, the number of branches increases by an addition of

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N

, which depends on the amount of time,

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\newline

When Aleksandra began the study, her tree had

52

\newline

Write a function that models the number of branches

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N(t)=\square

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Question

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is an element which loses

\frac{1}{10}

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M

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t

\newline

We measure that the initial mass of a sample of carbon

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960

\newline

Write a function that models the mass of the carbon-

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sample remaining

t

years since the initial measurement.

\newline

M(t)=

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