Identify the parent function

Oct 13, 2025 by magicalceylontours.com 41 views

Understanding the Function $f(x) = -4^{x+1} + 3$

In mathematics, functions are used to describe the relationship between variables. The given function $f(x) = -4^{x+1} + 3$ is a type of exponential function, which is characterized by its growth or decay pattern. In this case, we are dealing with a decay function, as indicated by the negative sign in front of the exponential term.

Growth or Decay (Parent Function)

As mentioned earlier, the parent function of the given function is a decay function. This means that as the input value of $x$ increases, the output value of $f(x)$ will decrease. The negative sign in front of the exponential term is responsible for this decay behavior.

Domain

The domain of a function is the set of all possible input values for which the function is defined. In the case of the given function, the domain is all real numbers. This means that the function is defined for any value of $x$ , and there are no restrictions on the input values.

Asymptote

An asymptote is a horizontal or vertical line that the graph of a function approaches as the input value of $x$ increases or decreases without bound. In the case of the given function, the asymptote is $y = 3$ . This means that as the input value of $x$ increases or decreases without bound, the output value of $f(x)$ will approach 3.

Range

The range of a function is the set of all possible output values for which the function is defined. In the case of the given function, the range is $y \ \textless \ 3$ . This means that the output value of $f(x)$ will always be less than 3.

Transformations

Transformations refer to the changes that are made to a function to obtain a new function. In the case of the given function, there are several transformations that have been applied:

Reflection across the x-axis: The negative sign in front of the exponential term causes the function to be reflected across the x-axis.
Vertical shift: The constant term $+3$ causes the function to be shifted vertically upwards by 3 units.
Horizontal shift: The term $x+1$ inside the exponential function causes the function to be shifted horizontally to the left by 1 unit.

Step-by-Step Solution

To determine the growth or decay, domain, asymptote, range, and transformations of the function $f(x) = -4^{x+1} + 3$ , we can follow these steps:

Identify the parent function: The parent function of the given function is a decay function, as indicated by the negative sign in front of the exponential term.
Determine the domain: The domain of the function is all real numbers, as there are no restrictions on the input values.
Find the asymptote: The asymptote of the function is $y = 3$ , as the output value of $f(x)$ will approach 3 as the input value of $x$ increases or decreases without bound.
Determine the range: The range of the function is $y \ \textless \ 3$ , as the output value of $f(x)$ will always be less than 3.
Identify the transformations: The function has undergone several transformations, including reflection across the x-axis, vertical shift, and horizontal shift.

Conclusion

Example Problems

Here are some example problems that illustrate the concepts discussed in this article:

Problem 1: Determine the growth or decay, domain, asymptote, range, and transformations of the function $f(x) = 2^{x-2} + 1$ .
Problem 2: Find the value of $f(x)$ when $x = 2$ for the function $f(x) = -3^{x+2} + 4$ .
Problem 3: Determine the equation of the asymptote of the function $f(x) = 2^{x+1} - 2$ .

Solutions

Here are the solutions to the example problems:

Problem 1: The function $f(x) = 2^{x-2} + 1$ is a growth function with a domain of all real numbers, an asymptote of $y = 1$ , a range of $y \ \textgreater \ 1$ , and several transformations, including reflection across the x-axis, vertical shift, and horizontal shift.
Problem 2: The value of $f(x)$ when $x = 2$ for the function $f(x) = -3^{x+2} + 4$ is $f(2) = -3^{2+2} + 4 = -81 + 4 = -77$ .
Problem 3: The equation of the asymptote of the function $f(x) = 2^{x+1} - 2$ is $y = -2$ .

Final Thoughts

In conclusion, the function $f(x) = -4^{x+1} + 3$ is a decay function with a domain of all real numbers, an asymptote of $y = 3$ , a range of $y \ \textless \ 3$ , and several transformations, including reflection across the x-axis, vertical shift, and horizontal shift. The concepts discussed in this article can be applied to a wide range of functions and problems, and are essential for understanding the behavior of functions in mathematics.
Q&A: Understanding the Function $f(x) = -4^{x+1} + 3$

In the previous article, we discussed the function $f(x) = -4^{x+1} + 3$ and its various properties, including growth or decay, domain, asymptote, range, and transformations. In this article, we will answer some frequently asked questions (FAQs) related to this function.

Q: What is the parent function of $f(x) = -4^{x+1} + 3$ ?

A: The parent function of $f(x) = -4^{x+1} + 3$ is a decay function, as indicated by the negative sign in front of the exponential term.

Q: What is the domain of $f(x) = -4^{x+1} + 3$ ?

A: The domain of $f(x) = -4^{x+1} + 3$ is all real numbers, as there are no restrictions on the input values.

Q: What is the asymptote of $f(x) = -4^{x+1} + 3$ ?

A: The asymptote of $f(x) = -4^{x+1} + 3$ is $y = 3$ , as the output value of $f(x)$ will approach 3 as the input value of $x$ increases or decreases without bound.

Q: What is the range of $f(x) = -4^{x+1} + 3$ ?

A: The range of $f(x) = -4^{x+1} + 3$ is $y \ \textless \ 3$ , as the output value of $f(x)$ will always be less than 3.

Q: What are the transformations of $f(x) = -4^{x+1} + 3$ ?

A: The function $f(x) = -4^{x+1} + 3$ has undergone several transformations, including reflection across the x-axis, vertical shift, and horizontal shift.

Q: How do I determine the growth or decay of a function?

A: To determine the growth or decay of a function, you need to look at the sign in front of the exponential term. If the sign is positive, the function is a growth function. If the sign is negative, the function is a decay function.

Q: How do I find the asymptote of a function?

A: To find the asymptote of a function, you need to look at the horizontal shift and vertical shift of the function. The asymptote is the value that the function approaches as the input value of $x$ increases or decreases without bound.

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

A: To determine the range of a function, you need to look at the vertical shift and horizontal shift of the function. The range is the set of all possible output values for which the function is defined.

Q: What are some common transformations of functions?

A: Some common transformations of functions include reflection across the x-axis, vertical shift, and horizontal shift.

Q: How do I apply transformations to a function?

A: To apply transformations to a function, you need to follow these steps:

Reflection across the x-axis: Multiply the function by -1.
Vertical shift: Add or subtract a constant value to the function.
Horizontal shift: Add or subtract a constant value to the input value of $x$ .

Q: What are some real-world applications of functions?

A: Functions have many real-world applications, including:

Modeling population growth: Functions can be used to model the growth of a population over time.
Modeling financial transactions: Functions can be used to model the value of a financial investment over time.
Modeling physical systems: Functions can be used to model the behavior of physical systems, such as the motion of an object under the influence of gravity.

Conclusion