Let’s solve each problem step by step. We’ll find the y-intercept, domain, range, and asymptote for each exponential function.
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Problem 3a: f(x) = 4(2)^x
Step 1: Find the y-intercept
The y-intercept is when x = 0.
f(0) = 4(2)^0 = 4(1) = 4 → So y-intercept is
4
Step 2: Domain
Exponential functions are defined for all real numbers → Domain:
all real numbers or (-∞, ∞)
Step 3: Range
Since 2^x is always positive, and we multiply by 4 (positive), f(x) > 0 → Range:
(0, ∞)
Step 4: Asymptote
As x → -∞, 2^x → 0, so f(x) → 0 → Horizontal asymptote at
y = 0
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Problem 3b: y = -4(2)^x
Step 1: y-intercept
x = 0 → y = -4(2)^0 = -4(1) = -4 → y-intercept:
-4
Step 2: Domain
Still all real numbers →
(-∞, ∞)
Step 3: Range
2^x > 0, but multiplied by -4 → y < 0 → Range:
(-∞, 0)
Step 4: Asymptote
As x → -∞, 2^x → 0, so y → 0 → Asymptote:
y = 0
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Problem 4a: y = 3^x + 5
Step 1: y-intercept
x = 0 → y = 3^0 + 5 = 1 + 5 = 6 → y-intercept:
6
Step 2: Domain
All real numbers →
(-∞, ∞)
Step 3: Range
3^x > 0 → 3^x + 5 > 5 → Range:
(5, ∞)
Step 4: Asymptote
As x → -∞, 3^x → 0 → y → 5 → Asymptote:
y = 5
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Problem 4b: f(x) = (1/3)^x - 5
Note: (1/3)^x is same as 3^(-x), still an exponential decay function.
Step 1: y-intercept
x = 0 → f(0) = (1/3)^0 - 5 = 1 - 5 = -4 → y-intercept:
-4
Step 2: Domain
All real numbers →
(-∞, ∞)
Step 3: Range
(1/3)^x > 0 → f(x) > -5 → Range:
(-5, ∞)
Step 4: Asymptote
As x → ∞, (1/3)^x → 0 → f(x) → -5 → Asymptote:
y = -5
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Final Answer:
3a)
a) y-intercept: 4
b) Domain: All real numbers
c) Range: y > 0
d) Asymptote: y = 0
3b)
a) y-intercept: -4
b) Domain: All real numbers
c) Range: y < 0
d) Asymptote: y = 0
4a)
a) y-intercept: 6
b) Domain: All real numbers
c) Range: y > 5
d) Asymptote: y = 5
4b)
a) y-intercept: -4
b) Domain: All real numbers
c) Range: y > -5
d) Asymptote: y = -5
Parent Tip: Review the logic above to help your child master the concept of exponential function worksheet pdf.
