In the graph you've described, Function #1 is linear (a straight dotted line) and Function #2 is exponential (an increasing solid curve).
The growth rates of the two functions compare as follows:
-
Linear Function (Function #1): A linear function grows at a constant rate. This means that for a given increase in the input (x-value), the output (y-value) increases by the same fixed amount. For example, if the function is defined as \( f(x) = mx + b \), the rate of change (slope) is constant at \( m \).
-
Exponential Function (Function #2): An exponential function grows at an increasing rate. This means that as the input (x-value) increases, the output (y-value) increases more rapidly. For instance, if the function is defined as \( g(x) = a \cdot b^x \) where \( b > 1 \), the growth accelerates as \( x \) increases.
In summary, while Function #1 grows at a constant pace (linear growth), Function #2 grows at an accelerating pace (exponential growth). Therefore, the growth rate of Function #2 will eventually surpass that of Function #1 as the input increases. Thus, the exponential function outgrows the linear function significantly as values become large enough.