Question

An exponential function with a base of 1/2 has been compressed vertically by a factor of 3/4 and reflected in the y-axis. Its asymptote is the line y=−4, and its intercept is (0,−13/4)(0,−13/4). Write an equation of the function and discuss its domain and range.

a) f(x)=−4×(1/2)(3/4)x, Domain: R, Range: (−[infinity],−4)(−[infinity],−4)
b) f(x)=−4×(2)−(4/3)x, Domain: R, Range: (−[infinity],−4)(−[infinity],−4)
c) f(x)=−4×(1/2)−(4/3)x, Domain: R, Range: (−4,[infinity])(−4,[infinity])
d) f(x)=−4×(2)(3/4)x, Domain: R, Range: (−4,[infinity])(−4,[infinity])

Answer 1

The correct exponential function equation meeting the described transformations is f(x) = -¾(1/2)-x - 4, with a domain of all real numbers and a range of (-4, infinity).

Step-by-step explanation:

The exponential function described has been vertically compressed by a factor of 3/4 and reflected in the y-axis. The function's asymptote is y = -4, and it passes through the point (0, -13/4). Taking into account these transformations, the correct equation for the function is f(x) = -¾(1/2)-x - 4. This is because the reflection in the y-axis changes the base from 1/2 to its reciprocal, which is 2, and the vertical compression is represented by the factor 3/4. However, this must be adjusted to account for the vertical asymptote by subtracting 4 from the entire function.

The domain of an exponential function is all real numbers (ℝ), as the function is defined for every real number x. The range of this function is (-4, ∞) because the function approaches but never reaches the horizontal asymptote y = -4 and extends to positive infinity for large negative values of x.