Which of the following is an exponential function?Group of answer choices𝑓(𝑥)=4⋅3𝑥+8𝑓(𝑥)=3⋅2𝑥5+4𝑓(𝑥)=3𝑥−2𝑓(𝑥)=2𝑥2−5𝑓(𝑥)=4𝑥3+2𝑥+5

To identify which of the given functions is an exponential function, we look for the mathematical form of an exponential function, which is generally expressed as $f(x) = a \cdot b^x$ where $a$ and $b$ are constants and $b > 0$.

Analyzing the Given Functions

1. Function 1: $f(x) = 4 \cdot 3^{x} + 8$

   • This function has an exponential term $3^{x}$, but the additional $+ 8$ makes it not a pure exponential function.

2. Function 2: $f(x) = 3 \cdot 2^{x/5} + 4$

   • This function has $2^{x/5}$ as an exponential component, but the $+ 4$ again makes it not a pure exponential function.

3. Function 3: $f(x) = 3^{x} - 2$

   • This has $3^{x}$, which is an exponential term. The $- 2$ does not affect the nature of being a pure exponential function.

4. Function 4: $f(x) = 2^{x^2} - 5$

   • This has $2^{x^2}$ which is also not in the pure exponential form as the exponent is quadratic.

5. Function 5: $f(x) = 4x^3 + 2x + 5$

   • This is a polynomial function and does not contain any exponential components.

Conclusion

According to the standard definition of an exponential function, the appropriate candidate from the available options is Function 3 $f(x) = 3^{x} - 2$, as it includes an exponential term, despite the constant subtracted from it.

Final Answer

The exponential function is $f(x) = 3^{x} - 2$.