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Sagot :
Let's find the composition of functions given [tex]\( f(x) = \sqrt[3]{x-4} \)[/tex] and [tex]\( g(x) = x^3 + 4 \)[/tex].
### (a) [tex]\( f \circ g \)[/tex]
The composition [tex]\( f \circ g \)[/tex] is defined as [tex]\( f(g(x)) \)[/tex].
1. Start with [tex]\( g(x) = x^3 + 4 \)[/tex].
2. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(g(x)) = f(x^3 + 4) \][/tex]
3. Now, use the definition of [tex]\( f(x) \)[/tex] which is [tex]\( f(x) = \sqrt[3]{x - 4} \)[/tex]:
[tex]\[ f(x^3 + 4) = \sqrt[3]{(x^3 + 4) - 4} \][/tex]
4. Simplify the expression inside the cube root:
[tex]\[ f(x^3 + 4) = \sqrt[3]{x^3} \][/tex]
5. The cube root of [tex]\( x^3 \)[/tex] is [tex]\( x \)[/tex]:
[tex]\[ f(x^3 + 4) = x \][/tex]
So,
[tex]\[ f \circ g = x \][/tex]
### Testing [tex]\( f \circ g \)[/tex] at [tex]\( x = 2 \)[/tex]:
[tex]\[ f \circ g (2) = 2 \][/tex]
Thus,
[tex]\[ f \circ g (2) = 2.0 \][/tex]
### (b) [tex]\( g \circ f \)[/tex]
The composition [tex]\( g \circ f \)[/tex] is defined as [tex]\( g(f(x)) \)[/tex].
1. Start with [tex]\( f(x) = \sqrt[3]{x-4} \)[/tex].
2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[ g(f(x)) = g(\sqrt[3]{x-4}) \][/tex]
3. Now, use the definition of [tex]\( g(x) \)[/tex] which is [tex]\( g(x) = x^3 + 4 \)[/tex]:
[tex]\[ g(\sqrt[3]{x-4}) = (\sqrt[3]{x-4})^3 + 4 \][/tex]
4. Simplify the expression:
[tex]\[ (\sqrt[3]{x-4})^3 = x - 4 \][/tex]
5. Thus,
[tex]\[ g(\sqrt[3]{x-4}) = (x - 4) + 4 \][/tex]
6. Simplify the expression:
[tex]\[ g(\sqrt[3]{x-4}) = x \][/tex]
So,
[tex]\[ g \circ f = x \][/tex]
### Testing [tex]\( g \circ f \)[/tex] at [tex]\( x = 2 \)[/tex]:
[tex]\[ g \circ f (2) = g(\sqrt[3]{2 - 4}) = g(\sqrt[3]{-2}) \][/tex]
[tex]\[ g(\sqrt[3]{-2}) = (\sqrt[3]{-2})^3 + 4 = -2 + 4 = 2 \][/tex]
[tex]\[ g \circ f (2) = 2 + 6.661338147750939 \times 10^{-16} \imath \][/tex]
Thus,
[tex]\[ g \circ f (2) = 2 + 6.661338147750939 \times 10^{-16} \imath \][/tex]
### (c) [tex]\( g \circ g \)[/tex]
The composition [tex]\( g \circ g \)[/tex] is defined as [tex]\( g(g(x)) \)[/tex].
1. Start with [tex]\( g(x) = x^3 + 4 \)[/tex].
2. Substitute [tex]\( g(x) \)[/tex] into itself:
[tex]\[ g(g(x)) = g(x^3 + 4) \][/tex]
3. Now, use the definition of [tex]\( g(x) \)[/tex]:
[tex]\[ g(x^3 + 4) = (x^3 + 4)^3 + 4 \][/tex]
4. Simplify inside the expression:
When evaluated at [tex]\( x = 2 \)[/tex]:
[tex]\[ g(x) = 2^3 + 4 = 8 + 4 = 12 \][/tex]
[tex]\[ g(g(2)) = g(12) = 12^3 + 4 = 1728 + 4 = 1732 \][/tex]
So,
[tex]\[ g \circ g = 1732 \][/tex]
Therefore, the compositions are:
[tex]\[ \begin{aligned} (a) & \quad f \circ g (2) = 2.0 \\ (b) & \quad g \circ f (2) = 2 + 6.661338147750939 \times 10^{-16} \imath \\ (c) & \quad g \circ g (2) = 1732 \\ \end{aligned} \][/tex]
### (a) [tex]\( f \circ g \)[/tex]
The composition [tex]\( f \circ g \)[/tex] is defined as [tex]\( f(g(x)) \)[/tex].
1. Start with [tex]\( g(x) = x^3 + 4 \)[/tex].
2. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(g(x)) = f(x^3 + 4) \][/tex]
3. Now, use the definition of [tex]\( f(x) \)[/tex] which is [tex]\( f(x) = \sqrt[3]{x - 4} \)[/tex]:
[tex]\[ f(x^3 + 4) = \sqrt[3]{(x^3 + 4) - 4} \][/tex]
4. Simplify the expression inside the cube root:
[tex]\[ f(x^3 + 4) = \sqrt[3]{x^3} \][/tex]
5. The cube root of [tex]\( x^3 \)[/tex] is [tex]\( x \)[/tex]:
[tex]\[ f(x^3 + 4) = x \][/tex]
So,
[tex]\[ f \circ g = x \][/tex]
### Testing [tex]\( f \circ g \)[/tex] at [tex]\( x = 2 \)[/tex]:
[tex]\[ f \circ g (2) = 2 \][/tex]
Thus,
[tex]\[ f \circ g (2) = 2.0 \][/tex]
### (b) [tex]\( g \circ f \)[/tex]
The composition [tex]\( g \circ f \)[/tex] is defined as [tex]\( g(f(x)) \)[/tex].
1. Start with [tex]\( f(x) = \sqrt[3]{x-4} \)[/tex].
2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[ g(f(x)) = g(\sqrt[3]{x-4}) \][/tex]
3. Now, use the definition of [tex]\( g(x) \)[/tex] which is [tex]\( g(x) = x^3 + 4 \)[/tex]:
[tex]\[ g(\sqrt[3]{x-4}) = (\sqrt[3]{x-4})^3 + 4 \][/tex]
4. Simplify the expression:
[tex]\[ (\sqrt[3]{x-4})^3 = x - 4 \][/tex]
5. Thus,
[tex]\[ g(\sqrt[3]{x-4}) = (x - 4) + 4 \][/tex]
6. Simplify the expression:
[tex]\[ g(\sqrt[3]{x-4}) = x \][/tex]
So,
[tex]\[ g \circ f = x \][/tex]
### Testing [tex]\( g \circ f \)[/tex] at [tex]\( x = 2 \)[/tex]:
[tex]\[ g \circ f (2) = g(\sqrt[3]{2 - 4}) = g(\sqrt[3]{-2}) \][/tex]
[tex]\[ g(\sqrt[3]{-2}) = (\sqrt[3]{-2})^3 + 4 = -2 + 4 = 2 \][/tex]
[tex]\[ g \circ f (2) = 2 + 6.661338147750939 \times 10^{-16} \imath \][/tex]
Thus,
[tex]\[ g \circ f (2) = 2 + 6.661338147750939 \times 10^{-16} \imath \][/tex]
### (c) [tex]\( g \circ g \)[/tex]
The composition [tex]\( g \circ g \)[/tex] is defined as [tex]\( g(g(x)) \)[/tex].
1. Start with [tex]\( g(x) = x^3 + 4 \)[/tex].
2. Substitute [tex]\( g(x) \)[/tex] into itself:
[tex]\[ g(g(x)) = g(x^3 + 4) \][/tex]
3. Now, use the definition of [tex]\( g(x) \)[/tex]:
[tex]\[ g(x^3 + 4) = (x^3 + 4)^3 + 4 \][/tex]
4. Simplify inside the expression:
When evaluated at [tex]\( x = 2 \)[/tex]:
[tex]\[ g(x) = 2^3 + 4 = 8 + 4 = 12 \][/tex]
[tex]\[ g(g(2)) = g(12) = 12^3 + 4 = 1728 + 4 = 1732 \][/tex]
So,
[tex]\[ g \circ g = 1732 \][/tex]
Therefore, the compositions are:
[tex]\[ \begin{aligned} (a) & \quad f \circ g (2) = 2.0 \\ (b) & \quad g \circ f (2) = 2 + 6.661338147750939 \times 10^{-16} \imath \\ (c) & \quad g \circ g (2) = 1732 \\ \end{aligned} \][/tex]
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