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For the real-valued functions g(x)=(x-4)/(x-5) and h(x)=4x+1 . find the composition gcirc h and specify its domain using interval notation. (gcirc h)(x)=square Domain of gcdot h:square

Pergunta

For the real-valued functions g(x)=(x-4)/(x-5) and h(x)=4x+1 . find the composition
gcirc h and specify its domain using interval notation.
(gcirc h)(x)=square 
Domain of gcdot h:square

For the real-valued functions g(x)=(x-4)/(x-5) and h(x)=4x+1 . find the composition gcirc h and specify its domain using interval notation. (gcirc h)(x)=square Domain of gcdot h:square

Solução

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PriscilaElite · Tutor por 8 anos

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To find the composition (g \circ h)(x)
, we need to substitute h(x)
into g(x)
.

Given:
g(x) = \frac{x-4}{x-5}

h(x) = 4x+1


Substituting h(x)
into g(x)
:
(g \circ h)(x) = g(h(x)) = g(4x+1) = \frac{(4x+1)-4}{(4x+1)-5} = \frac{4x-3}{4x-4}


The domain of (g \circ h)(x)
is the set of all real numbers except for the values that make the denominator zero. In this case, the denominator is (4x-4)
, which is zero when x = 1
. Therefore, the domain of (g \circ h)(x)
is all real numbers except x = 1
.

In interval notation, the domain of (g \circ h)(x)
is (-\infty, 1) \cup (1, \infty)
.
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