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Hunter-Best [27]
2 years ago
9

Show algebraically that f and g are inverse functions: g = sqrt(x+8)-4 f=x^2+8x+8

Mathematics
2 answers:
Tema [17]2 years ago
8 0

Answer:

See Below.

Step-by-step explanation:

By definition, if two functions, <em>f</em> and <em>g</em>, are inverses, then:

\displaystyle f(g(x)) = g(f(x)) = x

We are given the two functions:

\displaystyle g(x) = \sqrt{x + 8} -4 \text{ and } f(x) = x^2 + 8x + 8

Find f(g(x)) and g(f(x)):

\displaystyle \begin{aligned} f(g(x)) &= (\sqrt{x + 8} -4)^2 + 8(\sqrt{ x+ 8}-4) + 8 \\ \\ &= ((x+8) -8\sqrt{x + 8} +16) + 8\sqrt{x + 8} - 32 + 8 \\ \\ &= x + (-8\sqrt{x+8} + 8\sqrt{x + 8}) +(16 - 32 + 8 + 8) \\ \\ &= x \stackrel{\checkmark}{=} x\end{aligned}

And:

\displaystyle \begin{aligned}g(f(x)) &= \sqrt{(x^2 + 8x + 8) + 8} - 4 \\ \\ &= \sqrt{x^2 + 8x+16 } - 4 \\ \\ &=\sqrt{(x+ 4)^2} - 4 \\ \\ &= (x  +4) - 4 \\ \\ &= x \stackrel{\checkmark}{=} x  \end{aligned}

Hence, <em>f</em> and <em>g</em> are indeed inverses of each other.

Black_prince [1.1K]2 years ago
3 0

Answer:

f(x) and g(x) are not inverse functions.

Step-by-step explanation:

The results of the calculations in the attached image are not equal.

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