Understanding the Function Definitions

Oct 13, 2025 by thesweetdesigns.com 9 views

The given function $f$ is a piecewise function defined as:

$f(x)=\left\{\begin{array}{ll} 2^x, & x\ \textless \ 0 \\ -x^2-4x+1, & 0\ \textless \ x\ \textless \ 2 \\ \frac{1}{2}x+3, & x\ \textgreater \ 2 \end{array}\right.$

This function has three different definitions based on the value of $x$ . To evaluate the function, we need to determine which definition to use based on the value of $x$ .

Understanding the Function Definitions

The function $f$ has three different definitions:

For $x < 0$ : $f(x) = 2^x$
For $0 < x < 2$ : $f(x) = -x^2 - 4x + 1$
For $x > 2$ : $f(x) = \frac{1}{2}x + 3$

Evaluating the Function at Specific Values

To evaluate the function at specific values, we need to determine which definition to use based on the value of $x$ .

Evaluating the Function at $x = -1$

For $x = -1$ , we have $x < 0$ . Therefore, we use the first definition: $f(x) = 2^x$ . Substituting $x = -1$ , we get:

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

Evaluating the Function at $x = 1$

For $x = 1$ , we have $0 < x < 2$ . Therefore, we use the second definition: $f(x) = -x^2 - 4x + 1$ . Substituting $x = 1$ , we get:

$f(1) = -(1)^2 - 4(1) + 1 = -4$

Evaluating the Function at $x = 3$

For $x = 3$ , we have $x > 2$ . Therefore, we use the third definition: $f(x) = \frac{1}{2}x + 3$ . Substituting $x = 3$ , we get:

$f(3) = \frac{1}{2}(3) + 3 = \frac{9}{2}$

Conclusion

In conclusion, the function $f$ is a piecewise function defined as:

$f(x)=\left\{\begin{array}{ll} 2^x, & x\ \textless \ 0 \\ -x^2-4x+1, & 0\ \textless \ x\ \textless \ 2 \\ \frac{1}{2}x+3, & x\ \textgreater \ 2 \end{array}\right.$

To evaluate the function at specific values, we need to determine which definition to use based on the value of $x$ . The function has three different definitions based on the value of $x$ .

Key Takeaways

The function $f$ is a piecewise function defined as $f(x)=\left\{\begin{array}{ll} 2^x, & x\ \textless \ 0 \\ -x^2-4x+1, & 0\ \textless \ x\ \textless \ 2 \\ \frac{1}{2}x+3, & x\ \textgreater \ 2 \end{array}\right.$
To evaluate the function at specific values, we need to determine which definition to use based on the value of $x$ .
The function has three different definitions based on the value of $x$ .

Final Answer

Q: What is the function f(x)?

A: The function $f$ is a piecewise function defined as:

$f(x)=\left\{\begin{array}{ll} 2^x, & x\ \textless \ 0 \\ -x^2-4x+1, & 0\ \textless \ x\ \textless \ 2 \\ \frac{1}{2}x+3, & x\ \textgreater \ 2 \end{array}\right.$

Q: How do I evaluate the function f(x) at specific values?

A: To evaluate the function at specific values, you need to determine which definition to use based on the value of $x$ . If $x < 0$ , use the first definition: $f(x) = 2^x$ . If $0 < x < 2$ , use the second definition: $f(x) = -x^2 - 4x + 1$ . If $x > 2$ , use the third definition: $f(x) = \frac{1}{2}x + 3$ .

Q: What is the value of f(-1)?

A: For $x = -1$ , we have $x < 0$ . Therefore, we use the first definition: $f(x) = 2^x$ . Substituting $x = -1$ , we get:

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

Q: What is the value of f(1)?

A: For $x = 1$ , we have $0 < x < 2$ . Therefore, we use the second definition: $f(x) = -x^2 - 4x + 1$ . Substituting $x = 1$ , we get:

$f(1) = -(1)^2 - 4(1) + 1 = -4$

Q: What is the value of f(3)?

A: For $x = 3$ , we have $x > 2$ . Therefore, we use the third definition: $f(x) = \frac{1}{2}x + 3$ . Substituting $x = 3$ , we get:

$f(3) = \frac{1}{2}(3) + 3 = \frac{9}{2}$

Q: How do I determine which definition to use for a given value of x?

A: To determine which definition to use, you need to compare the value of $x$ with the given intervals. If $x < 0$ , use the first definition. If $0 < x < 2$ , use the second definition. If $x > 2$ , use the third definition.

Q: What are the key takeaways from this article?

A: The key takeaways from this article are:

The function $f$ is a piecewise function defined as $f(x)=\left\{\begin{array}{ll} 2^x, & x\ \textless \ 0 \\ -x^2-4x+1, & 0\ \textless \ x\ \textless \ 2 \\ \frac{1}{2}x+3, & x\ \textgreater \ 2 \end{array}\right.$
To evaluate the function at specific values, you need to determine which definition to use based on the value of $x$ .
The function has three different definitions based on the value of $x$ .

Q: What is the final answer to this problem?

A: The final answer is not provided as the problem is to evaluate the function at specific values and determine which statement is correct. However, the key takeaways and the conclusion provide a summary of the function and its evaluation.