Check the function for any restrictions

Oct 13, 2025 by leolavanderia.com.br 67 views

Graphing and Analyzing the Function $f(x) = 3^{x-5} + 1$

Understanding the Function

The given function is $f(x) = 3^{x-5} + 1$ . This is an exponential function with a base of 3, and it has been shifted 5 units to the right. The addition of 1 to the function shifts the entire graph up by 1 unit.

Domain and Range

The domain of a function is the set of all possible input values for which the function is defined. In the case of the exponential function $f(x) = 3^{x-5} + 1$ , the domain is all real numbers, or $(-\infty, \infty)$ . This is because the base of the exponent, 3, is a positive number, and the exponent $x-5$ can take on any real value.

The range of a function is the set of all possible output values for which the function is defined. For the function $f(x) = 3^{x-5} + 1$ , the range is also all real numbers, or $(-\infty, \infty)$ . This is because the function is an exponential function with a positive base, and the addition of 1 to the function shifts the entire graph up by 1 unit.

End Behavior

The end behavior of a function refers to the behavior of the function as the input values approach positive or negative infinity. For the function $f(x) = 3^{x-5} + 1$ , we can analyze the end behavior by looking at the exponent $x-5$ .

As $x \rightarrow \infty$ , the exponent $x-5$ approaches infinity, and the function $f(x) = 3^{x-5} + 1$ approaches infinity as well. This is because the base of the exponent, 3, is a positive number, and the exponent is increasing without bound.

As $x \rightarrow -\infty$ , the exponent $x-5$ approaches negative infinity, and the function $f(x) = 3^{x-5} + 1$ approaches 0. This is because the base of the exponent, 3, is a positive number, and the exponent is decreasing without bound.

Key Characteristics

The key characteristics of the function $f(x) = 3^{x-5} + 1$ are:

Domain: All real numbers, or $(-\infty, \infty)$
Range: All real numbers, or $(-\infty, \infty)$
End Behavior: As $x \rightarrow \infty$ , $f(x) \rightarrow \infty$ , and as $x \rightarrow -\infty$ , $f(x) \rightarrow 0$

Graphing the Function

To graph the function $f(x) = 3^{x-5} + 1$ , we can use the following steps:

Plot the y-intercept: The y-intercept of the function is the point where the graph intersects the y-axis. To find the y-intercept, we can plug in $x=0$ into the function: $f(0) = 3^{0-5} + 1 = 3^{-5} + 1 = \frac{1}{243} + 1 = \frac{244}{243}$ .
Plot the x-intercept: The x-intercept of the function is the point where the graph intersects the x-axis. To find the x-intercept, we can plug in $y=0$ into the function: $0 = 3^{x-5} + 1$ . Solving for $x$ , we get $x-5 = -\infty$ , so $x = -\infty$ .
Plot the asymptotes: The asymptotes of the function are the lines that the graph approaches as the input values approach positive or negative infinity. The asymptotes of the function are the lines $y=0$ and $x=-\infty$ .
Plot the graph: Using the points and asymptotes we have plotted, we can sketch the graph of the function.

Conclusion

In conclusion, the function $f(x) = 3^{x-5} + 1$ is an exponential function with a base of 3, and it has been shifted 5 units to the right. The domain and range of the function are all real numbers, or $(-\infty, \infty)$ . The end behavior of the function is that as $x \rightarrow \infty$ , $f(x) \rightarrow \infty$ , and as $x \rightarrow -\infty$ , $f(x) \rightarrow 0$ . The key characteristics of the function are its domain, range, and end behavior.
Q&A: Graphing and Analyzing the Function $f(x) = 3^{x-5} + 1$

Frequently Asked Questions

Q: What is the domain of the function $f(x) = 3^{x-5} + 1$ ?

A: The domain of the function $f(x) = 3^{x-5} + 1$ is all real numbers, or $(-\infty, \infty)$ . This is because the base of the exponent, 3, is a positive number, and the exponent $x-5$ can take on any real value.

Q: What is the range of the function $f(x) = 3^{x-5} + 1$ ?

A: The range of the function $f(x) = 3^{x-5} + 1$ is also all real numbers, or $(-\infty, \infty)$ . This is because the function is an exponential function with a positive base, and the addition of 1 to the function shifts the entire graph up by 1 unit.

Q: What is the end behavior of the function $f(x) = 3^{x-5} + 1$ ?

A: The end behavior of the function $f(x) = 3^{x-5} + 1$ is that as $x \rightarrow \infty$ , $f(x) \rightarrow \infty$ , and as $x \rightarrow -\infty$ , $f(x) \rightarrow 0$ . This is because the base of the exponent, 3, is a positive number, and the exponent is increasing or decreasing without bound.

Q: How do I graph the function $f(x) = 3^{x-5} + 1$ ?

A: To graph the function $f(x) = 3^{x-5} + 1$ , you can use the following steps:

Plot the y-intercept: The y-intercept of the function is the point where the graph intersects the y-axis. To find the y-intercept, plug in $x=0$ into the function.
Plot the x-intercept: The x-intercept of the function is the point where the graph intersects the x-axis. To find the x-intercept, plug in $y=0$ into the function.
Plot the asymptotes: The asymptotes of the function are the lines that the graph approaches as the input values approach positive or negative infinity. The asymptotes of the function are the lines $y=0$ and $x=-\infty$ .
Plot the graph: Using the points and asymptotes you have plotted, sketch the graph of the function.

Q: What are the key characteristics of the function $f(x) = 3^{x-5} + 1$ ?

A: The key characteristics of the function $f(x) = 3^{x-5} + 1$ are:

Domain: All real numbers, or $(-\infty, \infty)$
Range: All real numbers, or $(-\infty, \infty)$
End Behavior: As $x \rightarrow \infty$ , $f(x) \rightarrow \infty$ , and as $x \rightarrow -\infty$ , $f(x) \rightarrow 0$

Q: How do I determine the domain and range of a function?

A: To determine the domain and range of a function, you can use the following steps:

Check the function for any restrictions: Check the function for any restrictions on the input values, such as division by zero or taking the square root of a negative number.
Determine the domain: The domain of the function is the set of all possible input values for which the function is defined.
Determine the range: The range of the function is the set of all possible output values for which the function is defined.

Q: How do I determine the end behavior of a function?

A: To determine the end behavior of a function, you can use the following steps:

Check the function for any asymptotes: Check the function for any asymptotes, such as vertical or horizontal asymptotes.
Determine the end behavior: The end behavior of the function is the behavior of the function as the input values approach positive or negative infinity.

Conclusion