Question

which graph represents a function with an initial value of \\(\frac{1}{2}\\)?

Explanation:

Step1: Understand Initial Value

The initial value of a function is the value when \( x = 0 \) (the y - intercept). We need to find the graph where the y - intercept (value at \( x = 0 \)) is \( \frac{1}{2}=0.5 \).

Step2: Analyze the Given Graph

In the first graph, the point at \( x = 0 \) is \( (0, 0.4) \), which is close to but not equal to \( 0.5 \). Wait, maybe there is another graph (not fully shown here, but let's recall the concept). The initial value is \( f(0) \). So we need to check which graph has \( f(0)=\frac{1}{2} \). If we assume there are other graphs, for example, if a graph passes through \( (0, \frac{1}{2}) \), that would be the one. But from the given first graph, the y - intercept is \( 0.4 \), but maybe the other graph (not shown completely) has a y - intercept of \( 0.5 \). Let's think about exponential functions. The general form of an exponential function is \( y = ab^x \), where \( a \) is the initial value (when \( x = 0 \), \( y=a \)). So we need \( a=\frac{1}{2} \), so the graph where when \( x = 0 \), \( y=\frac{1}{2} \) is the correct one.

(Note: Since the full set of graphs is not shown, but based on the concept, the graph with y - intercept \( \frac{1}{2} \) (i.e., the point \( (0, \frac{1}{2}) \) on the graph) is the answer. If we consider the first graph, the y - intercept is \( 0.4 \), so maybe the other graph (the one not fully displayed) has \( (0, 0.5) \). Let's assume that the correct graph is the one where at \( x = 0 \), \( y=\frac{1}{2} \).)

Answer:

The graph with a y - intercept (value at \( x = 0 \)) of \( \frac{1}{2} \) (i.e., the point \( (0, \frac{1}{2}) \) lies on the graph) is the one representing the function with an initial value of \( \frac{1}{2} \). (If we had to choose from the given partial graph and others, the one with \( f(0)=\frac{1}{2} \) is correct.)