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Art [367]
3 years ago
15

(a) Find the domain off (x) = ln(ex − 3). (b) Find F −1 and its domain.

Mathematics
1 answer:
Anettt [7]3 years ago
5 0

Answer:

a) The domain of f(x) is x > 1.1.

b)

The inverse function is:

y = \ln{(e^{x} + 3)}The domain is all the real values of x.Step-by-step explanation:(a) Find the domain off f(x) = ln(e^x − 3)The domain of f(x) = ln(g(x)) is g(x) > 0. That means that the ln function only exists for positive values.So, here we have[tex]g(x) = e^{x} - 3

So we need

e^{x} - 3 > 0

e^{x} > 3

Applying ln to both sides

\ln{e^{x}} > \ln{3}

x > 1.1

So the domain of f(x) is x > 1.1.

(b) Find F −1 and its domain.

F^{-1} is the inverse function of f.

How do we find the inverse function?

To find the inverse equation, we change y with x to form the new equation, and then we isolate y in the new equation. So:

Original equation:

f(x) = y = \ln{e^{x} - 3}

New equation

x = \ln{e^{y} - 3}

Here, we apply the exponential to both sides:

e^{x} = e^{\ln{e^{y} - 3}}

e^{y} - 3 = e^{x}

e^{y} = e^{x} + 3

Applying ln to both sides

\ln{e^{y}} = \ln{e^{x} + 3}

The inverse function is:

y = \ln{e^{x} + 3}

The domain is

e^{x} + 3 > 0

e^{x} > -3

e^{x} is always a positive number, so it is always going to be larger than -3 no matter the value of x. So the domain are all the real values.

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