Final answer:
The function f(x) = ab^cx, where b > 0 and b != 1, is continuous and one-to-one with a domain of all real numbers. The range is positive if a > 0 and negative if a < 0. It is increasing with a positive c and decreasing with a negative c, the horizontal asymptote is y = 0, and the y-intercept is at (0, a).
Step-by-step explanation:
The function given is f(x) = abcx, where b > 0 and b ≠ 1. Here's a breakdown of the function's properties:
- Continuity: Since the function is an exponential function and b is a positive real number not equal to 1, f(x) is continuous for all real numbers.
- One-to-one: If c is positive, f(x) is an increasing function, and it is one-to-one. If c is negative, f(x) is decreasing and still one-to-one.
- Domain: The domain of f(x) is all real numbers since there are no restrictions on x in the function definition.
- Range: If a > 0, the range is all positive real numbers (0, +∞). If a < 0, the range is all negative real numbers (-∞, 0).
- Increasing/Decreasing: f(x) is increasing if the value of c is positive, and f(x) is decreasing if the value of c is negative.
- Horizontal Asymptote: The equation of the horizontal asymptote is y = 0.
- Y-Intercept: The coordinates of the y-intercept are (0, abc·0) which simplifies to (0, a).