Question
Which of the following is an exponential function? \( y=3^{x} \quad y=2 x^{3} \quad y=x^{1 / 2} \)
Ask by Clarke Boyd. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( y = 3^{x} \) is the exponential function.
Solution
Among the given functions:
1. **\( y = 3^{x} \)**
This is an **exponential function** because the variable \( x \) is in the exponent. It follows the general form \( y = a \cdot b^{x} \), where \( a = 1 \) and \( b = 3 \).
2. **\( y = 2x^{3} \)**
This is a **polynomial function** where \( x \) is raised to the power of 3.
3. **\( y = x^{1/2} \)**
This is a **radical function** (specifically, the square root of \( x \)).
**Conclusion:**
The exponential function among the options provided is **\( y = 3^{x} \)**.
**Answer:**
Only the first function, \( y = 3^{\,x} \), is an exponential function.
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Simplify this solution Bonus Knowledge
The function \( y = 3^{x} \) is an exponential function because it has the form \( y = a^{x} \), where \( a \) is a positive constant (in this case, 3) and the variable \( x \) is in the exponent. This means that the base is being raised to the power of the variable, leading to rapid growth or decay depending on the base's value. In contrast, the other two functions, \( y = 2x^{3} \) and \( y = x^{1/2} \), are polynomial functions. They do not exhibit the unique properties of exponential functions, such as constant percentage growth. Remember, exponential growth is like a rocket taking off, while polynomial functions are more like a steady ascent—you’ll see differences in their curves!
