Which of these is an
exponential function?

Ask by Chadwick West. in the United States

Jan 24,2025

Question

Which of these is an exponential function? $$ y=x^{2}+\frac{1}{2} \quad y=\left(\frac{1}{4}\right)^{x} $$

UpStudy AI · Accepted Answer

An **exponential function** is characterized by having the variable in the exponent. The general form of an exponential function is:

$$ y = a \cdot b^x $$

where:
- $$ a $$ is a constant,
- $$ b $$ is the base (with $$ b > 0 $$ and $$ b \neq 1 $$),
- $$ x $$ is the exponent.

Let's analyze the given functions:

1. **$$ y = x^2 + \frac{1}{2} $$**  
   - Here, $$ x $$ is the base raised to the power of 2, and then a constant is added.  
   - This is a **polynomial function**, not an exponential function, because the variable $$ x $$ is not in the exponent.

2. **$$ y = \left(\frac{1}{4}\right)^x $$**  
   - In this case, $$ \frac{1}{4} $$ is the base, and $$ x $$ is in the exponent.  
   - This fits the form of an exponential function perfectly.

**Conclusion:**  
The function $$ y = \left(\frac{1}{4}\right)^x $$ is an exponential function.

**Answer:**  
Only the second function, $$ y = \left(\dfrac{1}{4}\right)^{\,x} $$, is an exponential function.
The second function, $$ y = \left(\frac{1}{4}\right)^x $$, is an exponential function.

Which of these is an
exponential function?

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