Calculating the Value of the Function

Oct 13, 2025 by miodeya.com 12 views

Step 1: Find the Value of $x$ that Makes the Exponent Zero

To find the value of $x$ that makes the exponent zero, we need to solve the equation $x-1=0$ . This equation can be solved by adding 1 to both sides of the equation.

Solving the Equation

$x - 1 = 0$

Adding 1 to both sides of the equation:

$x - 1 + 1 = 0 + 1$

$x = 1$

Calculating the Value of the Function

Now that we have found the value of $x$ , we can calculate the value of the function $f(x) = 2^{x-1}$ by substituting $x = 1$ into the function.

$f(1) = 2^{1-1}$

$f(1) = 2^0$

$f(1) = 1$

Understanding the Graph

The graph of the function $f(x) = 2^{x-1}$ is an exponential function that passes through the point $(1, 1)$ . The graph of an exponential function is a curve that increases rapidly as the value of $x$ increases.

Key Features of the Graph

The graph passes through the point $(1, 1)$ .
The graph is an exponential function that increases rapidly as the value of $x$ increases.
The graph has a horizontal asymptote at $y = 0$ .

Graphing the Function

To graph the function $f(x) = 2^{x-1}$ , we can use a graphing calculator or a computer algebra system. The graph of the function is a curve that passes through the point $(1, 1)$ and increases rapidly as the value of $x$ increases.

Interpreting the Graph

The graph of the function $f(x) = 2^{x-1}$ can be interpreted in various ways. For example, the graph can be used to model population growth, where the value of $x$ represents time and the value of $f(x)$ represents the population size.

Real-World Applications

The graph of the function $f(x) = 2^{x-1}$ has various real-world applications. For example, the graph can be used to model population growth, where the value of $x$ represents time and the value of $f(x)$ represents the population size.

Conclusion

In conclusion, the graph of the function $f(x) = 2^{x-1}$ is an exponential function that passes through the point $(1, 1)$ . The graph of the function increases rapidly as the value of $x$ increases and has a horizontal asymptote at $y = 0$ . The graph of the function has various real-world applications, including modeling population growth.

Step 2: Find the Value of $x$ that Makes the Exponent Negative

To find the value of $x$ that makes the exponent negative, we need to solve the equation $x - 1 < 0$ . This equation can be solved by adding 1 to both sides of the equation.

Solving the Equation

$x - 1 < 0$

Adding 1 to both sides of the equation:

$x - 1 + 1 < 0 + 1$

$x < 1$

Calculating the Value of the Function

Now that we have found the value of $x$ , we can calculate the value of the function $f(x) = 2^{x-1}$ by substituting $x < 1$ into the function.

$f(x) = 2^{x-1}$

Since $x < 1$ , we know that $x - 1 < 0$ . Therefore, we can conclude that $f(x) < 1$ .

Understanding the Graph

The graph of the function $f(x) = 2^{x-1}$ is an exponential function that decreases rapidly as the value of $x$ decreases.

Key Features of the Graph

The graph decreases rapidly as the value of $x$ decreases.
The graph has a horizontal asymptote at $y = 0$ .

Graphing the Function

To graph the function $f(x) = 2^{x-1}$ , we can use a graphing calculator or a computer algebra system. The graph of the function is a curve that decreases rapidly as the value of $x$ decreases.

Interpreting the Graph

The graph of the function $f(x) = 2^{x-1}$ can be interpreted in various ways. For example, the graph can be used to model population decline, where the value of $x$ represents time and the value of $f(x)$ represents the population size.

Real-World Applications

The graph of the function $f(x) = 2^{x-1}$ has various real-world applications. For example, the graph can be used to model population decline, where the value of $x$ represents time and the value of $f(x)$ represents the population size.

Conclusion

In conclusion, the graph of the function $f(x) = 2^{x-1}$ is an exponential function that decreases rapidly as the value of $x$ decreases. The graph of the function has a horizontal asymptote at $y = 0$ and has various real-world applications, including modeling population decline.

Step 3: Find the Value of $x$ that Makes the Exponent Positive

To find the value of $x$ that makes the exponent positive, we need to solve the equation $x - 1 > 0$ . This equation can be solved by adding 1 to both sides of the equation.

Solving the Equation

$x - 1 > 0$

Adding 1 to both sides of the equation:

$x - 1 + 1 > 0 + 1$

$x > 1$

Calculating the Value of the Function

Now that we have found the value of $x$ , we can calculate the value of the function $f(x) = 2^{x-1}$ by substituting $x > 1$ into the function.

$f(x) = 2^{x-1}$

Since $x > 1$ , we know that $x - 1 > 0$ . Therefore, we can conclude that $f(x) > 1$ .

Understanding the Graph

The graph of the function $f(x) = 2^{x-1}$ is an exponential function that increases rapidly as the value of $x$ increases.

Key Features of the Graph

The graph increases rapidly as the value of $x$ increases.
The graph has a horizontal asymptote at $y = 0$ .

Graphing the Function

To graph the function $f(x) = 2^{x-1}$ , we can use a graphing calculator or a computer algebra system. The graph of the function is a curve that increases rapidly as the value of $x$ increases.

Interpreting the Graph

Real-World Applications

Conclusion

In conclusion, the graph of the function $f(x) = 2^{x-1}$ is an exponential function that increases rapidly as the value of $x$ increases. The graph of the function has a horizontal asymptote at $y = 0$ and has various real-world applications, including modeling population growth.

Step 4: Find the Value of $x$ that Makes the Exponent Zero

To find the value of $x$ that makes the exponent zero, we need to solve the equation $x - 1 = 0$ . This equation can be solved by adding 1 to both sides of the equation.

Solving the Equation

$x - 1 = 0$

Adding 1 to both sides of the equation:

$x - 1 + 1 = 0 + 1$

$x = 1$

Calculating the Value of the Function

Now that we have found the value of $x$ , we can calculate the value of the function $f(x) = 2^{x-1}$ by substituting $x = 1$ into the function.

$f(1) = 2^{1-1}$

$f(1) = 2^0$

$f(1) = 1$

Understanding the Graph

The graph of the function $f(x) = 2^{x-1}$ is an exponential function that passes through the point $(1, 1)$ .

Key Features of the Graph

The graph passes through the point $(1, 1)$ .
The graph is an exponential function that increases rapidly as the value of $x$ increases.
The graph has a horizontal asymptote at $y = 0$ .

Graphing the Function

$To graph the function $f(x) = 2^{x-1}$, we can use a graphing calculator or a computer algebra system. The graph of the function is a curve that<br/> **Graph: $f(x)=2^{x-1}$ Q&A**$

Q: What is the value of $x$ that makes the exponent zero?

A: To find the value of $x$ that makes the exponent zero, we need to solve the equation $x - 1 = 0$ . This equation can be solved by adding 1 to both sides of the equation.

Solving the Equation

$x - 1 = 0$

Adding 1 to both sides of the equation:

$x - 1 + 1 = 0 + 1$

$x = 1$

Q: What is the value of the function $f(x)$ when $x = 1$ ?

A: Now that we have found the value of $x$ , we can calculate the value of the function $f(x) = 2^{x-1}$ by substituting $x = 1$ into the function.

$f(1) = 2^{1-1}$

$f(1) = 2^0$

$f(1) = 1$

Q: What is the key feature of the graph of the function $f(x) = 2^{x-1}$ ?

A: The key feature of the graph of the function $f(x) = 2^{x-1}$ is that it is an exponential function that increases rapidly as the value of $x$ increases.

Q: What is the horizontal asymptote of the graph of the function $f(x) = 2^{x-1}$ ?

A: The horizontal asymptote of the graph of the function $f(x) = 2^{x-1}$ is $y = 0$ .

Q: How can the graph of the function $f(x) = 2^{x-1}$ be used in real-world applications?

A: The graph of the function $f(x) = 2^{x-1}$ can be used in real-world applications such as modeling population growth, where the value of $x$ represents time and the value of $f(x)$ represents the population size.

Q: What is the value of $x$ that makes the exponent negative?

A: To find the value of $x$ that makes the exponent negative, we need to solve the equation $x - 1 < 0$ . This equation can be solved by adding 1 to both sides of the equation.

Solving the Equation

$x - 1 < 0$

Adding 1 to both sides of the equation:

$x - 1 + 1 < 0 + 1$

$x < 1$

Q: What is the value of the function $f(x)$ when $x < 1$ ?

A: Now that we have found the value of $x$ , we can calculate the value of the function $f(x) = 2^{x-1}$ by substituting $x < 1$ into the function.

$f(x) = 2^{x-1}$

Since $x < 1$ , we know that $x - 1 < 0$ . Therefore, we can conclude that $f(x) < 1$ .

Q: What is the value of $x$ that makes the exponent positive?

A: To find the value of $x$ that makes the exponent positive, we need to solve the equation $x - 1 > 0$ . This equation can be solved by adding 1 to both sides of the equation.

Solving the Equation

$x - 1 > 0$

Adding 1 to both sides of the equation:

$x - 1 + 1 > 0 + 1$

$x > 1$

Q: What is the value of the function $f(x)$ when $x > 1$ ?

A: Now that we have found the value of $x$ , we can calculate the value of the function $f(x) = 2^{x-1}$ by substituting $x > 1$ into the function.

$f(x) = 2^{x-1}$

Since $x > 1$ , we know that $x - 1 > 0$ . Therefore, we can conclude that $f(x) > 1$ .

Q: What is the key feature of the graph of the function $f(x) = 2^{x-1}$ when $x > 1$ ?

A: The key feature of the graph of the function $f(x) = 2^{x-1}$ when $x > 1$ is that it is an exponential function that increases rapidly as the value of $x$ increases.

Q: What is the horizontal asymptote of the graph of the function $f(x) = 2^{x-1}$ when $x > 1$ ?

A: The horizontal asymptote of the graph of the function $f(x) = 2^{x-1}$ when $x > 1$ is $y = 0$ .