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Using the functions ݂
f(x)=-3x and
h(x)=(1)/(x+1), find (
h
f (
(x)). State the domain of ℎ ∘ ݂
f in interval notation.

User Simpanoz
by
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1 Answer

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Answer:


\textsf{Domain:} \quad \left(-\infty, (1)/(3)\right) \cup \left((1)/(3), \infty\right)


\textsf{Range:} \quad \left(-\infty, 0\right) \cup \left(0, \infty\right)

Explanation:

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

The range of a function is the set of all possible output values (y-values) for which the function is defined.

Given functions:


f(x)=-3x


h(x)=(1)/(x+1)

First, determine the composite function (h o f)(x) by substituting function f(x) in place of the x in function h(x):


\begin{aligned}(h \circ f)(x)&=h(-3x)\\\\&=(1)/((-3x)+1)\\\\&=(1)/(-3x+1)\end{aligned}

As function (h o f)(x) is a rational function, it is undefined when its denominator equals zero:


\begin{aligned}-3x+1&=0\\\\-3x&=-1\\\\x&=(1)/(3)\end{aligned}

So, the composite function is undefined when x = 1/3.

Therefore, its domain is:


\left(-\infty, (1)/(3)\right) \cup \left((1)/(3), \infty\right)

As the degree of the numerator of (h o f)(x) is less than the degree of its denominator, a horizontal asymptote occurs at y = 0. Therefore, the range of the composite function is restricted to:


\left(-\infty, 0\right) \cup \left(0, \infty\right)

User Ameya Pandilwar
by
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