Question

the graph below could be the graph of which exponential function?
a. $f(x) = 3 \cdot (1.2)^x$
b. $f(x) = 3^x$
c. $f(x) = 3 \cdot (-1.2)^x$
d. $f(x) = 3 \cdot (0.2)^x$

Explanation:

Step1: Analyze the general form of exponential functions

The general form of an exponential function is \( f(x) = a \cdot b^x \), where \( a \) is the initial value (y - intercept when \( x = 0 \)) and \( b \) determines the growth or decay. If \( b>1 \), it is an exponential growth function; if \( 0 < b < 1 \), it is an exponential decay function; and \( b\) cannot be negative because for real - valued functions, \( b^x \) is not real for all real \( x \) when \( b<0 \).

Step2: Analyze the y - intercept

For the function \( f(x)=a\cdot b^x \), when \( x = 0 \), \( f(0)=a\cdot b^0=a \). From the graph, when \( x = 0 \), the y - intercept is \( 3 \) (since the graph crosses the y - axis at \( y = 3 \)). So \( a = 3 \).

Step3: Analyze the type of exponential function (growth or decay)

The graph is increasing as \( x \) increases, which means it is an exponential growth function. For an exponential growth function, the base \( b>1 \).

Step4: Analyze each option

• Option A: \( f(x)=3\cdot(1.2)^x \). Here, \( a = 3 \) (matches the y - intercept) and \( b = 1.2>1 \) (so it is an exponential growth function).
• Option B: \( f(x)=3^x \). When \( x = 0 \), \( f(0)=3^0 = 1

eq3 \), so it does not match the y - intercept of the graph.

• Option C: \( f(x)=3\cdot(- 1.2)^x \). Since the base \( b=-1.2<0 \), the function is not a real - valued function for all real \( x \) (for example, when \( x=\frac{1}{2} \), \( (-1.2)^{\frac{1}{2}}=\sqrt{- 1.2} \) is not real), so this function is not valid for the graph of a real - valued function.
• Option D: \( f(x)=3\cdot(0.2)^x \). Here, \( b = 0.2<1 \), so it is an exponential decay function, but the graph is increasing (exponential growth), so this option is incorrect.

Answer:

A. \( F(x)=3\cdot(1.2)^x \)