Find the domain and range for the following exponential function.

a) Record the horizontal asymptote below. Be sure to record you answer as .
b) Record the domain below using interval notation.
c) Record the range below using interval notation.

Ask by Gray Daniels. in the United States

Nov 02,2024

Question

Find the domain and range for the following exponential function.

a) Record the horizontal asymptote below. Be sure to record you answer as .
b) Record the domain below using interval notation.
c) Record the range below using interval notation.

Ask by Gray Daniels. in the United States

Nov 02,2024

UpStudy AI · Accepted Answer

To analyze the function $$ f(x) = -e^{x-5} + 8 $$, we can determine the domain, range, and horizontal asymptote.

### a) Horizontal Asymptote
The function $$ f(x) = -e^{x-5} + 8 $$ has a horizontal asymptote determined by the behavior of the exponential term as $$ x $$ approaches infinity.

As $$ x 	o \infty $$:
- $$ e^{x-5} 	o \infty $$
- Therefore, $$ -e^{x-5} 	o -\infty $$
- Thus, $$ f(x) 	o -\infty + 8 = -\infty $$

As $$ x 	o -\infty $$:
- $$ e^{x-5} 	o 0 $$
- Therefore, $$ -e^{x-5} 	o 0 $$
- Thus, $$ f(x) 	o 0 + 8 = 8 $$

The horizontal asymptote is at $$ y = 8 $$.

**Answer:** $$ y = 8 $$

### b) Domain
The domain of the function $$ f(x) = -e^{x-5} + 8 $$ is all real numbers since the exponential function is defined for all real values of $$ x $$.

**Answer:** $$ (-\infty, \infty) $$

### c) Range
To find the range, we analyze the output values of the function.

From our earlier analysis:
- As $$ x 	o -\infty $$, $$ f(x) 	o 8 $$.
- As $$ x 	o \infty $$, $$ f(x) 	o -\infty $$.

The function approaches 8 but never actually reaches it (since $$ e^{x-5} $$ is always positive). Therefore, the function can take any value less than 8.

Thus, the range of $$ f(x) $$ is all values less than 8.

**Answer:** $$ (-\infty, 8) $$

### Summary of Answers
a) $$ y = 8 $$  
b) Domain: $$ (-\infty, \infty) $$  
c) Range: $$ (-\infty, 8) $$
a) $$ y = 8 $$ b) Domain: $$ (-\infty, \infty) $$ c) Range: $$ (-\infty, 8) $$

Find the domain and range for the following exponential function.

a) Record the horizontal asymptote below. Be sure to record you answer as .
b) Record the domain below using interval notation.
c) Record the range below using interval notation.

Upstudy AI Solution

Answer

Solution

Mind Expander

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Find the domain and range for the following exponential function. a) Record the horizontal asymptote below. Be sure to record you answer as . b) Record the domain below using interval notation. c) Record the range below using interval notation.

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Find the domain and range for the following exponential function.

a) Record the horizontal asymptote below. Be sure to record you answer as .
b) Record the domain below using interval notation.
c) Record the range below using interval notation.