How do you find $(f @ g)(x)$ and its domain, $(g @ f)(x)$ and its domain, $(f @ g)(-2)$ and $(g @ f)(-2)$ of the following problem $f(x) = x+ 2$ , $g(x) = 2x^2$ ?

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Answer 1 · 2018-01-17T13:33:25

See the explanation below...

By the composition $(f\circ g)(x)$ , we mean $f(g(x))$ and by $(g\circ f)(x)$ , we mean $g(f(x))$ .

To find $f(g(x))$ , we need to put the value of $g(x)$ for every value of $x$ in $f(x)$ . So, by doing this, we get:

$(f\circ g)(x)=f(g(x))=(2x^2)+2$

$=2x^2+2$

The domain of a function is the set of values for which the function is real and defined.

This above evaluated function has no undefined points. The domain is $-oo < x < oo$

Similarly, we have:

$(g\circ f)(x)=g(f(x))=2(x+2)^2$

Simplify:

$=2x^2+8x+8$

Domain: $-oo < x < oo$

Now, in the same manner, find $(f\circ g)(-2)$ as:

$=2(-2)^2+2$

Simplify:

$=10$

And:

$(g\circ f)(-2)=2(-2)^2+8(-2)+8$

Simplify:

$=0$

How do you find (f @ g)(x)(f @ g)(x) and its domain, (g @ f)(x)(g @ f)(x) and its domain, (f @ g)(-2) (f @ g)(-2) and (g @ f)(-2)(g @ f)(-2) of the following problem f(x) = x+ 2f(x) = x+ 2, g(x) = 2x^2g(x) = 2x^2?