Question

For each exponential function below (a-f), answer the following:

i. State the domain and range
ii. Determine the y-intercept
iii. State the equation of the horizontal asymptote
iv. Sketch the function

a) (y = 3ˣ)

b) (y = 0.25ˣ)

c) (y = -(2ˣ))

d) (y = 2(0.3)ˣ)

e) (y = 2(0.3)ˣ)

f) (y = 4(0.5)ˣ + 5)

Answer 1

The domain, range, y-intercept, equation of horizontal asymptote, and sketch for each exponential function are provided in a step-by-step manner.

Step-by-step explanation:

a) y = 3ˣ

i. Domain: All real numbers, Range: (0, +∞)
ii. Y-intercept: (0, 1)
iii. Horizontal asymptote: y = 0
iv. Sketch: The graph starts at (0, 1) and increases rapidly as x increases.

b) y = 0.25ˣ

i. Domain: All real numbers, Range: (0, +∞)
ii. Y-intercept: (0, 1)
iii. Horizontal asymptote: y = 0
iv. Sketch: The graph starts at (0, 1) and decreases rapidly as x increases.

c) y = -(2ˣ)

i. Domain: All real numbers, Range: (-∞, 0)
ii. Y-intercept: (0, -1)
iii. Horizontal asymptote: y = 0
iv. Sketch: The graph starts at (0, -1) and decreases rapidly as x increases.

d) y = 2(0.3)ˣ

i. Domain: All real numbers, Range: (0, +∞)
ii. Y-intercept: (0, 2)
iii. Horizontal asymptote: y = 0
iv. Sketch: The graph starts at (0, 2) and decreases as x increases.

e) y = 2(0.3)ˣ

i. Domain: All real numbers, Range: (0, +∞)
ii. Y-intercept: (0, 2)
iii. Horizontal asymptote: y = 0
iv. Sketch: The graph starts at (0, 2) and decreases as x increases.

f) y = 4(0.5)ˣ + 5

i. Domain: All real numbers, Range: (5, +∞)
ii. Y-intercept: (0, 9)
iii. Horizontal asymptote: y = 5
iv. Sketch: The graph starts at (0, 9), increases as x increases, and approaches a horizontal line at y = 5.