Final answer:
The exponential function that does not have an x-intercept is f(x) = 5^x-5 - 1, as the function will always be positive even after subtracting 1, ensuring it never crosses the x-axis.
Step-by-step explanation:
The student is looking for an exponential function that does not have an x-intercept. An x-intercept occurs where the graph of the function crosses or touches the x-axis, meaning the value of the function f(x) at that point is 0. Generally, an exponential function has the form f(x) = abx + c, where a and b are constants, and c is the y-intercept.
For an exponential function not to have an x-intercept, the function must always be positive or always be negative, which means c must ensure that the function does not cross the x-axis. Given the options:
- f(x) = 5x-5 - 1
- f(x) = 5x-5 - 5
- f(x) = -5x-5 + 5
- f(x) = -5x-5 - 1
The correct answer would be the first one, because all exponential functions with positive base (like 5) are always positive, and subtracting 1 will still keep it above the x-axis, hence
f(x) = 5
x-5 - 1 does not have an x-intercept.