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Tems11 [23]
3 years ago
12

What's does tumultuously mean??

Mathematics
1 answer:
yuradex [85]3 years ago
8 0
<span>Making a loud, confused noise; uproarious.</span>
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<em>f(x)</em> = 1/(1 - <em>x</em>)

<em>g(x)</em> = (<em>x</em> - 1)/<em>x</em>

The inverse functions <em>f</em> ⁻¹<em>(x)</em> and <em>g </em>⁻¹<em>(x)</em>, if they exist, are such that

<em>f( f</em> ⁻¹<em>(x) )</em> = <em>x</em>

<em>g( g</em> ⁻¹<em>(x) )</em> = <em>x</em>

So we have

<em>f( f</em> ⁻¹<em>(x) )</em> = 1/(1 - <em>f</em> ⁻¹<em>(x) </em>) = <em>x</em>

Solve for <em>f</em> ⁻¹<em>(x)</em> :

1 = (1 - <em>f</em> ⁻¹<em>(x)</em> ) <em>x</em>

1 - <em>f</em> ⁻¹<em>(x)</em> = 1/<em>x</em>

<em>f</em> ⁻¹<em>(x)</em> = 1 - 1/<em>x</em>

<em>f</em> ⁻¹<em>(x)</em> = (<em>x</em> - 1)/<em>x</em>

and so <em>f</em> ⁻¹<em>(x)</em> = <em>g(x)</em>.

Similarly, you can solve for <em>g</em> ⁻¹<em>(x)</em> <em>:</em>

<em>g(</em> <em>g</em> ⁻¹<em>(x) )</em> = (<em>g</em> ⁻¹<em>(x)</em> - 1) / <em>g</em> ⁻¹<em>(x)</em> = <em>x</em>

1 - 1/<em>g</em> ⁻¹<em>(x)</em> = <em>x</em>

1/<em>g</em> ⁻¹<em>(x)</em> = 1 - <em>x</em>

<em>g</em> ⁻¹<em>(x)</em> = 1/(1 - <em>x</em>)

So we know <em>f(x)</em> and <em>g(x)</em> are inverses of one another,

<em>f</em> ⁻¹<em>(x)</em> = <em>g(x)</em>

<em>g</em> ⁻¹<em>(x)</em> = <em>f(x)</em>

Then

(a) (<em>f</em> o <em>g</em>)(<em>x</em>) = <em>x</em>

(b) (<em>g</em> o <em>f</em> ⁻¹)(<em>x</em>) = <em>g(g(x))</em> = (<em>g(x)</em> - 1)/<em>g(x)</em> = 1 - 1/<em>g(x)</em> = 1 - <em>x</em>/(<em>x</em> - 1) = 1/(1 - <em>x</em>)

(c) (<em>f</em> o<em> g</em> ⁻¹)(<em>x</em>) = <em>f(f(x))</em> = 1/(1 - <em>f(x) </em>) = 1/(1 - 1/(1 - <em>x</em>)) = 1/(<em>x</em>/(<em>x</em> - 1)) = (<em>x</em> - 1)/<em>x</em>

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Last summer, Betty earned $4200 by painting houses. She invested
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