Question

the linear function h(x) has a slope of 2 and passes through the points (0,6) and (-3,0). The exponential function k(x) follows a pattern of exponential growth and passes through the points (-2,0) and (0,1). on which interval are both functions positive?

Answer 1

The interval where both the linear function h(x) with a slope of 2 and the exponential growth function k(x) are positive is from x = 0 to infinity.

Step-by-step explanation:

We want to identify the interval where both the linear function h(x) and the exponential function k(x) are positive. Given that h(x) has a slope of 2 and passes through the points (0,6) and (-3,0), we can determine the equation of the linear function to be h(x) = 2x + 6. This function is positive for all x ≥5.

For the exponential growth function k(x) that passes through points (-2,0) and (0,1), we know that the general form of an exponential function is y = ab^x, where a is the initial amount and b is the growth factor. Plugging in the points (-2,0) and (0,1), we find that b>1 and a must be positive, and k(x) will be positive for all x greater than -2.

Given these functions, the interval where both are positive is from x = 0 to infinity, since h(x) is always positive from x = 0 and k(x) is positive for all x greater than -2, including from x = 0 onwards.