Question

Which of the following are exponential functions?
(1 point)
Responses

f left parenthesis x right parenthesis equals 10 superscript 4 plus 3 x baseline
Image with alt text: f left parenthesis x right parenthesis equals 10 superscript 4 plus 3 x baseline

f left parenthesis x right parenthesis equals left parenthesis negative 5 right parenthesis superscript x baseline
Image with alt text: f left parenthesis x right parenthesis equals left parenthesis negative 5 right parenthesis superscript x baseline

f left parenthesis x right parenthesis equals left parenthesis start fraction 1 over x end fraction right parenthesis superscript 6 baseline
Image with alt text: f left parenthesis x right parenthesis equals left parenthesis start fraction 1 over x end fraction right parenthesis superscript 6 baseline

f left parenthesis x right parenthesis equals left parenthesis one fourth right parenthesis superscript negative 2 x baseline
Image with alt text: f left parenthesis x right parenthesis equals left parenthesis one fourth right parenthesis superscript negative 2 x baseline

Answer 1

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:

1. **\( f(x) = 10^4 + 3x \)**:
- This is not an exponential function. It is a linear function with a constant term \( 10^4 \) and a linear term \( 3x \).

2. **\( f(x) = (-5)^x \)**:
- This is also not an exponential function because the base is negative. Exponential functions require a positive base.

3. **\( f(x) = \left(\frac{1}{x}\right)^6 \)**:
- This can be rewritten as \( f(x) = x^{-6} \), which is a power function and not an exponential function.

4. **\( f(x) = \left(\frac{1}{4}\right)^{-2x} \)**:
- 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:
- \( f(x) = \left(\frac{1}{4}\right)^{-2x} \)