Question - Determining the Function Type Based on the Exponential Expression

The image presented shows an exponential function of the form \( f(x) = 2 \cdot 5^x \). To discuss this function, we do not need to solve a specific question but we can analyze its properties.

Firstly, let's note that an exponential function has a constant base raised to a variable exponent. In this case, the base is 5 and the exponent is \( x \). The coefficient 2 in front of the exponential term scales the output of the function. This function represents exponential growth since the base (5) is greater than 1.

Here are some key features of the function \( f(x) = 2 \cdot 5^x \):

- Exponential Growth: Because the base of the exponent is greater than 1, the function will exhibit exponential growth, meaning it will increase rapidly as \( x \) gets larger.

- Y-intercept: The y-intercept of this function can be found by evaluating \( f(0) \). Since any non-zero number raised to the power of 0 is 1, \( f(0) = 2 \cdot 5^0 = 2 \cdot 1 = 2 \). Therefore, the y-intercept is at the point (0, 2).

- Asymptotic Behavior: As \( x \) approaches negative infinity, \( f(x) \) approaches 0, but never reaches it. This means the x-axis is a horizontal asymptote.

- Domain and Range: The domain of this function is all real numbers, since you can raise 5 to any real power. The range, however, is limited to positive values because an exponential function with a positive base always yields positive results when multiplied by a positive coefficient. Therefore, the range is \( (0, \infty) \).

If you were asked to graph this function, you'd notice that it crosses the y-axis at point (0, 2) and increases rapidly towards positive infinity as \( x \) increases. Additionally, as \( x \) decreases, the graph approaches the x-axis but never touches or crosses it.

Understanding these properties helps in sketching the graph of the function and in solving various applications involving exponential growth, such as compound interest problems, population growth models, and radioactive decay among others.