153k views
0 votes
Fill in the Blanks

Given, f(x) = ab^cx, b > 0, b ≠ 1.
Is f(x) continuous?
Is f(x) one-to-one?
Domain is _________.
If a > 0, range is _________, if a < 0, range is _________.
f(x) is increasing if the value of c is _________.
f(x) is decreasing if the value of c is _________.
Equation of the horizontal asymptote is _________.
Coordinates of y-intercept are _________.

User Anouar
by
8.8k points

1 Answer

4 votes

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).

User Slothrop
by
7.2k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories