To determine which of the functions are exponential functions, we need to look for functions of the form \( f(x) = a^x \), where \( a \) is a positive real number.
Let's analyze each given function:
-
\( f(x) = 10^4 + 3x \):
- This is not an exponential function. It is a linear function with a constant term \( 10^4 \) and a linear term \( 3x \).
-
\( f(x) = (-5)^x \):
- This is also not an exponential function because the base is negative. Exponential functions require a positive base.
-
\( f(x) = \left(\frac{1}{x}\right)^6 \):
- This can be rewritten as \( f(x) = x^{-6} \), which is a power function and not an exponential function.
-
\( f(x) = \left(\frac{1}{4}\right)^{-2x} \):
- This can be rewritten as \( f(x) = 4^{2x} \) (since \( \left(\frac{1}{4}\right)^{-1} = 4 \)). This is indeed an exponential function with a positive base.
Conclusion:
The only exponential function from the list is:
- \( f(x) = \left(\frac{1}{4}\right)^{-2x} \)