Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 12: Problem 56
Is the given function an exponential function? $$ F(x)=0.4^{x+1} $$
Short Answer
Expert verified
Yes, \(F(x) = 0.4^{x+1}\) is an exponential function.
Step by step solution
01
Identify the Base
In an exponential function, the base is a constant raised to the power of a variable or an expression in terms of a variable. Here, the base is clearly given as 0.4 in the expression \(0.4^{x+1}\).
02
Check for a Constant Base
An exponential function has the form \(a^x\), where \(a\) is a constant. In \(F(x) = 0.4^{x+1}\), the base 0.4 is a constant, a characteristic of an exponential function.
03
Examine the Exponent
The exponent in an exponential function should be a variable or expression involving the variable. In \(F(x) = 0.4^{x+1}\), the exponent is \(x+1\), which is indeed involving a variable \(x\).
04
Conclusion
Since the base is constant and the exponent involves a variable, \(F(x) = 0.4^{x+1}\) is indeed an exponential function.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Base of a function
In the study of functions, the term "base" refers to a specific component of an exponential function. When we express an exponential function like \(a^x\), the base is the number or constant represented by \(a\). This base is foundational because it defines how the function behaves and grows as the exponent changes.
- In the equation \(F(x) = 0.4^{x+1}\), the base is 0.4.
- The base determines how "steep" or "flat" the exponential curve is.
Constant base
A constant base in an exponential function means that the base doesn't change; it remains fixed throughout the function. This constant characteristic is a hallmark of exponential expressions, differentiating them from other mathematical expressions where the base might be variable.
When the base is constant, like in \(F(x) = 0.4^{x+1}\), you will always see the same base number raised to a power, which is crucial for classifying a function as exponential.
When the base is constant, like in \(F(x) = 0.4^{x+1}\), you will always see the same base number raised to a power, which is crucial for classifying a function as exponential.
- The constancy of the base simplifies calculations and predictions.
- It allows the function to have predictable behavior.
Variable exponent
In an exponential function, the exponent plays an essential role and is often a variable or an expression involving a variable. This variability is what gives exponential functions their unique properties and dynamic growth or decay characteristics.
- In \(F(x) = 0.4^{x+1}\), the exponent is \(x+1\).
- This exponent involves the variable \(x\), influencing the function's output directly based on \(x\)'s value.
Exponential expression
An exponential expression, usually presented in the form \(a^{b}\), is a mathematical statement involving a constant base raised to the power of a variable exponent. This type of expression is fundamental to understanding and identifying exponential functions.The expression \(F(x) = 0.4^{x+1}\) is an excellent example of an exponential expression, characterized by its constant base of 0.4 and a variable expression \(x+1\) as the exponent.
- Exponential expressions are known for their significant impact due to rapid growth (or decay).
- They are vital in fields like finance and natural sciences, modeling scenarios like compound interest and natural decay.
