A. Linear functions grow at a constant rate, meaning they increase by the same amount over equal intervals. For example, if a linear function increases by 2 every time (like y = 2x), it will go up by 2 for each step in x.
Exponential functions, on the other hand, grow by a multiplying factor. This means that as the value gets larger, the increments also get bigger. For example, in an exponential function like y = 2^x, every time x increases by 1, y doubles, which leads to much faster growth as x becomes larger.
B. The statement “No matter what the growth rate of a linear function is, that of an exponential function will always exceed it” means that as time goes on, exponential functions will eventually outgrow linear functions, regardless of the linear growth rate.
For example, if we have a linear function like y = 2x (growing by 2 as x increases) and an exponential function like y = 2^x (growing by doubling), at first, the linear function may seem competitive, but as x gets larger, the exponential function grows much faster. This can be seen in specific cases: for small values of x, 2x might be larger than 2^x, but as x increases to values like 10 or 20, 2^x becomes far larger than 2x, demonstrating that exponential growth will always outpace linear growth in the long run.
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