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Find [tex]f \circ g, g \circ f[/tex], and [tex]g \circ g[/tex].

Given:
[tex]\[ f(x) = x^2 \][/tex]
[tex]\[ g(x) = x - 6 \][/tex]

(a) [tex]f \circ g[/tex]
[tex]\[ \square \][/tex]

(b) [tex]g \circ f[/tex]
[tex]\[ \square \][/tex]

(c) [tex]g \circ g[/tex]
[tex]\[ \square \][/tex]

Sagot :

GinnyAnswer
Sure, let's find each of the compositions step-by-step:

(a) [tex]\( f \circ g \)[/tex]:

[tex]\( f \circ g(x) \)[/tex] means we apply [tex]\( g \)[/tex] first and then apply [tex]\( f \)[/tex] to the result.

Given [tex]\( g(x) = x - 6 \)[/tex]:

1. First apply [tex]\( g \)[/tex]:
[tex]\[ g(x) = x - 6 \][/tex]

2. Now substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(g(x)) = f(x - 6) \][/tex]

3. Since [tex]\( f(x) = x^2 \)[/tex], replace [tex]\( x \)[/tex] with [tex]\( x - 6 \)[/tex]:
[tex]\[ f(x - 6) = (x - 6)^2 \][/tex]

So,
[tex]\[ f \circ g(x) = (x - 6)^2 \][/tex]

(b) [tex]\( g \circ f \)[/tex]:

[tex]\( g \circ f(x) \)[/tex] means we apply [tex]\( f \)[/tex] first and then apply [tex]\( g \)[/tex] to the result.

Given [tex]\( f(x) = x^2 \)[/tex]:

1. First apply [tex]\( f \)[/tex]:
[tex]\[ f(x) = x^2 \][/tex]

2. Now substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[ g(f(x)) = g(x^2) \][/tex]

3. Since [tex]\( g(x) = x - 6 \)[/tex], replace [tex]\( x \)[/tex] with [tex]\( x^2 \)[/tex]:
[tex]\[ g(x^2) = x^2 - 6 \][/tex]

So,
[tex]\[ g \circ f(x) = x^2 - 6 \][/tex]

(c) [tex]\( g \circ g \)[/tex]:

[tex]\( g \circ g(x) \)[/tex] means we apply [tex]\( g \)[/tex] first and then apply [tex]\( g \)[/tex] again to the result.

Given [tex]\( g(x) = x - 6 \)[/tex]:

1. First apply [tex]\( g \)[/tex]:
[tex]\[ g(x) = x - 6 \][/tex]

2. Now substitute [tex]\( g(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[ g(g(x)) = g(x - 6) \][/tex]

3. Since [tex]\( g(x) = x - 6 \)[/tex], replace [tex]\( x \)[/tex] with [tex]\( x - 6 \)[/tex]:
[tex]\[ g(x - 6) = (x - 6) - 6 = x - 12 \][/tex]

So,
[tex]\[ g \circ g(x) = x - 12 \][/tex]

Therefore, the detailed solutions are:

(a) [tex]\( f \circ g(x) = (x - 6)^2 \)[/tex]

(b) [tex]\( g \circ f(x) = x^2 - 6 \)[/tex]

(c) [tex]\( g \circ g(x) = x - 12 \)[/tex]
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