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Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

f(x) = quantity x minus nine divided by quantity x plus five. and g(x) = quantity negative five x minus nine divided by quantity x minus one.
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Answer :

kudzordzifrancis

ANSWER

See below

EXPLANATION

Given

[tex]f(x) = \frac{ {x}- 9 }{x + 5} [/tex]

and

[tex]g(x) = \frac{ - 5x - 9}{x - 1} [/tex]

[tex](f \circ \: g)(x)= \frac{ (\frac{ - 5x - 9}{x - 1})- 9 }{(\frac{ - 5x - 9}{x - 1} )+ 5} [/tex]

[tex](f \circ \: g)(x)= \frac{ \frac{ - 5x - 9 - 9(x - 1)}{x - 1}}{\frac{ - 5x - 9 + 5(x - 1)}{x - 1} } [/tex]

Expand:

[tex](f \circ \: g)(x)= \frac{ \frac{ - 5x - 9 - 9x + 9}{x - 1}}{\frac{ - 5x - 9 + 5x - 5}{x - 1} } [/tex]

[tex](f \circ \: g)(x)= \frac{ \frac{ - 5x - 9x + 9 - 9}{x - 1}}{\frac{ - 5x + 5x - 5 - 9}{x - 1} } [/tex]

[tex](f \circ \: g)(x)= \frac{ \frac{ - 14x }{x - 1}}{\frac{ -14}{x - 1} } [/tex]

Since the denominators are the same, they will cancel out,

[tex](f \circ \: g)(x)= \frac{ - 14x}{ - 14} = x[/tex]

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