Question

The following values represent exponential function ƒ(x) and linear function g(x).

ƒ(1) = 2 g(1) = 2.5
ƒ(2) = 6 g(2) = 4
A. Determine whether or not there is a solution to the equation In 2-3 sentences describe whether there is a solution to the equation ƒ(x)=g(x) between x=1 and x=2.

B. Use complete sentences to justify your claim.

Answer 1

There is indeed a solution to the equation ƒ(x)=g(x) between x=1 and x=2. This is because the exponential function ƒ(x) starts below g(x) at x=1, but surpasses it by x=2, implying at least one intersection point between them due to continuous growth.

Step-by-step explanation:

To determine whether there is a solution to the equation ƒ(x)=g(x) between x=1 and x=2, let's first analyze the function values provided. We have ƒ(1) = 2 and g(1) = 2.5, as well as ƒ(2) = 6 and g(2) = 4. Observing that ƒ(x) is an exponential function, we can infer that its value grows rapidly, while g(x), being a linear function, increases in a constant manner. Therefore, ƒ(x) starts lower, crosses over g(x), and then exceeds it before x=2.

Based on this, we can conclude that there exists an x-value between 1 and 2 where the two functions are equal, since the exponential function is continuously increasing and has a value less than the linear function at x=1, but a greater value at x=2. This conclusion relies on the Intermediate Value Theorem, which states that if a function is continuous on a closed interval and takes on different signs at the ends of the interval, then the function must have a root within that interval.

Answer 2

There is a solution for f(x) = g(x) between x = 1 and x = 2.