Question

Graphing exponential functions and then State the domain, range and asymptote​

Answer 1

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Answer:

  D: all real numbers

  R: f(x) > 0

  A: f(x) = 0

  (-∞, 0), (+∞, +∞)

  vertical stretch by a factor of 2; left shift 2 units

Step-by-step explanation:

The transformation ...

  g(x) = a·f(b(x -c)) +d

does the following:

• vertical stretch by a factor of 'a'
• horizontal compression by a factor of 'b'
• translation right by 'c' units
• translation up by 'd' units

For many functions, horizontal coordinate changes are indistinguishable from vertical coordinate changes. Exponential functions tend to be one of those.

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Using the above notation, you seem to have f(x) = 3^x, and g(x) = 2f(x+2). The transformation is a vertical stretch by a factor of 2, and a translation left 2 units.

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As with all exponential functions, ...

• the domain is "all real numbers"
• the range is all numbers above the asymptote: f(x) > 0
• the horizontal asymptote is f(x) = 0

The function is a growth function, so ...

• x → -∞, f(x) → 0
• x → ∞, f(x) → ∞

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Additional comment

The left shift is equivalent to an additional vertical stretch. The function could be rewritten as ...

  f(x) = 18(3^x)

with no left shift and a vertical stretch by a factor of 18 instead of 2.