Analyzing the Components of the Function

Oct 13, 2025 by windailys.com 104 views

Understanding the Graph of the Function $f(x) = -(2)^x + 4$

The given function $f(x) = -(2)^x + 4$ is an exponential function with a negative coefficient. This means that as the value of $x$ increases, the value of $f(x)$ will decrease, and vice versa. The graph of this function will have a downward trend.

Analyzing the Components of the Function

To understand the behavior of the function, let's break it down into its components. The first part, $-(2)^x$ , is an exponential function with a base of 2 and a negative coefficient. This means that the function will decrease as $x$ increases. The second part, $+4$ , is a constant that shifts the graph of the function upwards.

Graphing the Function

To graph the function, we can start by plotting a few points. Let's choose some values of $x$ and calculate the corresponding values of $f(x)$ .

$x$	$f(x) = -(2)^x + 4$
-2	$-(2)^{-2} + 4 = -0.25 + 4 = 3.75$
-1	$-(2)^{-1} + 4 = -0.5 + 4 = 3.5$
0	$-(2)^0 + 4 = -1 + 4 = 3$
1	$-(2)^1 + 4 = -2 + 4 = 2$
2	$-(2)^2 + 4 = -4 + 4 = 0$

From these points, we can see that the graph of the function is a downward trend, with a minimum value of 0 at $x = 2$ .

Finding the Range of the Function

The range of a function is the set of all possible output values. In this case, we want to find the range of the function $f(x) = -(2)^x + 4$ . To do this, we need to consider the behavior of the function as $x$ approaches positive and negative infinity.

As $x$ approaches positive infinity, the value of $f(x)$ approaches $-\infty$ . This is because the exponential function $-(2)^x$ grows without bound as $x$ increases.

As $x$ approaches negative infinity, the value of $f(x)$ approaches $4$ . This is because the exponential function $-(2)^x$ approaches $0$ as $x$ decreases, and the constant $+4$ shifts the graph upwards.

Conclusion

Based on the analysis above, we can conclude that the range of the function $f(x) = -(2)^x + 4$ is $\{y \mid -\infty \ \textless \ y \ \textless \ 4\}$ . This means that the function can take on any value between $-\infty$ and $4$ , but not including $4$ .

Answer

The correct answer is B. $\{y \mid -\infty \ \textless \ y \ \textless \ 4\}$
Q&A: Understanding the Graph of the Function $f(x) = -(2)^x + 4$

Q: What is the range of the function $f(x) = -(2)^x + 4$ ?

A: The range of the function $f(x) = -(2)^x + 4$ is $\{y \mid -\infty \ \textless \ y \ \textless \ 4\}$ . This means that the function can take on any value between $-\infty$ and $4$ , but not including $4$ .

Q: Why does the function $f(x) = -(2)^x + 4$ have a downward trend?

A: The function $f(x) = -(2)^x + 4$ has a downward trend because the exponential function $-(2)^x$ grows without bound as $x$ increases, causing the value of $f(x)$ to decrease.

Q: What is the minimum value of the function $f(x) = -(2)^x + 4$ ?

A: The minimum value of the function $f(x) = -(2)^x + 4$ is $0$ , which occurs at $x = 2$ .

Q: How does the constant $+4$ affect the graph of the function $f(x) = -(2)^x + 4$ ?

A: The constant $+4$ shifts the graph of the function $f(x) = -(2)^x + 4$ upwards, causing the minimum value of the function to be $0$ instead of $-4$ .

Q: What happens to the value of $f(x)$ as $x$ approaches positive infinity?

A: As $x$ approaches positive infinity, the value of $f(x)$ approaches $-\infty$ because the exponential function $-(2)^x$ grows without bound as $x$ increases.

Q: What happens to the value of $f(x)$ as $x$ approaches negative infinity?

A: As $x$ approaches negative infinity, the value of $f(x)$ approaches $4$ because the exponential function $-(2)^x$ approaches $0$ as $x$ decreases, and the constant $+4$ shifts the graph upwards.

Q: Can the function $f(x) = -(2)^x + 4$ take on any value between $-\infty$ and $4$ ?

A: Yes, the function $f(x) = -(2)^x + 4$ can take on any value between $-\infty$ and $4$ , but not including $4$ .

Q: Is the function $f(x) = -(2)^x + 4$ continuous?

A: Yes, the function $f(x) = -(2)^x + 4$ is continuous because it is a polynomial function, and polynomial functions are continuous everywhere.

Q: Is the function $f(x) = -(2)^x + 4$ differentiable?

A: Yes, the function $f(x) = -(2)^x + 4$ is differentiable because it is a polynomial function, and polynomial functions are differentiable everywhere.

Q: Can the function $f(x) = -(2)^x + 4$ be used to model real-world phenomena?

A: Yes, the function $f(x) = -(2)^x + 4$ can be used to model real-world phenomena, such as population growth or decay, where the rate of change is exponential.