Final answer:
The domain of the function y = 5*(2^x) - 1 is all real numbers, the range is from -1 to infinity, and the horizontal asymptote is the line y = -1.
Step-by-step explanation:
State Domain, Range, and Asymptote
The question deals with the function y = 5*(2^x) - 1. To find the domain, range, and asymptote of this function:
Domain: The domain of an exponential function like 2^x is all real numbers. Therefore, the domain of y = 5*(2^x) - 1 is also all real numbers, or (-∞, ∞).
Range: Since 2^x is always positive and multiplied by 5, the smallest value of 5*(2^x) is 0, when x approaches negative infinity. Subtracting 1 means the smallest value of y is -1. Hence, the range is (-1, ∞).
Horizontal Asymptote: The horizontal asymptote occurs where the function levels off as x goes to negative infinity. In this case, y approaches -1, so the equation of the horizontal asymptote is y = -1.The given equation is y = 5*(2ˣ) - 1. We can determine the state domain, range, and asymptote of this equation with the following steps:
State Domain:
The domain of an exponential function is all real numbers. So, the domain of this equation is (-∞, ∞).
State Range:The range of an exponential function depends on the sign of the coefficient before the exponential term. In this case, the coefficient is positive, which means the range is (a, ∞), where a is the y-value of the y-intercept. To find the y-intercept, substitute x = 0 into the equation:
y = 5*(2⁰) - 1 = 5*1 - 1 = 4
Therefore, the range is (4, ∞).
State Asymptote:
An exponential function with a positive coefficient has a horizontal asymptote at y = 0. So, the equation y = 0 is the asymptote.