Given the following exponential functions:
Function 1: [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex]
Function 2: [tex]\( y = 3^{-x} \)[/tex]
We need to determine whether these two functions are equivalent.
Step-by-step solution:
1. Rewrite Function 1:
First, let's analyze Function 1 and see if we can rewrite it in another form. Function 1 is:
[tex]\[
y = \left(\frac{1}{3}\right)^x
\][/tex]
2. Rewrite the Fraction:
The fraction [tex]\(\frac{1}{3}\)[/tex] can be expressed using negative exponents. Specifically, [tex]\(\frac{1}{3}\)[/tex] is equivalent to [tex]\(3^{-1}\)[/tex]. Therefore, we can rewrite Function 1 as:
[tex]\[
y = \left(3^{-1}\right)^x
\][/tex]
3. Simplify the Exponentiation:
When we raise a power to another power, we multiply the exponents. So, [tex]\(\left(3^{-1}\right)^x\)[/tex] simplifies to:
[tex]\[
y = 3^{-1 \cdot x} = 3^{-x}
\][/tex]
4. Compare with Function 2:
Now, we compare the rewritten form of Function 1 ([tex]\(y = 3^{-x}\)[/tex]) with Function 2, which is also:
[tex]\[
y = 3^{-x}
\][/tex]
We observe that both functions are indeed identical.
Conclusion:
Function 1 [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex] and Function 2 [tex]\( y = 3^{-x} \)[/tex] are equivalent. They represent the same exponential relationship. Thus, the correct statement is that both functions are equivalent.
