For the following function, state:

i. the domain and range

ii. the y-intercept, and

iii. the horizontal asymptote

f(x)=(2)x−7

error i mean f(x)=(2)^x-7

domain is all reals

since 2^x > 0, 2^x-7 > -7, so the range is (-7,∞)

I expect you can do the intercepts, right? Just set x=0 or y=0.

To find the domain and range, y-intercept, and the horizontal asymptote of the given function f(x) = 2x - 7, we can analyze the function and look for some key characteristics.

i. Domain and Range:
The domain of a function refers to all the possible values of x for which the function is defined. In this case, the function does not involve any square roots, fractions with variables in the denominator, or any other mathematical operations that would lead to undefined values. Therefore, the domain is all real numbers (-∞, +∞).

The range of a function refers to all the possible values of y or f(x) that the function can yield. Since the function is a linear equation with the variable x raised to the power of 1 (i.e., degree 1), the range is also all real numbers (-∞, +∞).

ii. Y-Intercept:
To find the y-intercept, we need to evaluate the function when x = 0. Substituting x = 0 into the function, we get:

f(0) = 2(0) - 7
f(0) = -7

Therefore, the y-intercept is -7.

iii. Horizontal Asymptote:
For linear functions, there is no horizontal asymptote since the graph of a linear function extends indefinitely in both the positive and negative directions without approaching a particular value.

To summarize:

i. Domain: (-∞, +∞)
ii. Range: (-∞, +∞)
iii. Y-Intercept: -7
iv. Horizontal Asymptote: None