Question

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7.ƒ(x)=√(x+4),g(x)=−(2x)ƒ(x)=\surd(x+4),g(x)=-(\frac{2}{x})

Accepted Answer

Step 1: Understand the composition of functions. For (ƒ∘g)(x), this means ƒ(g(x)), which is the function ƒ applied to the output of g(x). Similarly, for (g∘ƒ)(x), this means g(ƒ(x)). Step 2: Write the expressions for each composition. For (ƒ∘g)(x), substitute g(x) into ƒ(x):

$$(ƒ∘g)(x) = ƒ(g(x)) = \sqrt{g(x) + 4}$$

Since $$g(x) = -\frac{2}{x}$$, this becomes:

$$(ƒ∘g)(x) = \sqrt{-\frac{2}{x} + 4}$$ Step 3: Determine the domain of (ƒ∘g)(x). The domain consists of all x-values for which the expression inside the square root is greater than or equal to zero (because the square root function requires a non-negative input), and where g(x) itself is defined (denominator not zero). So, solve the inequality:

$$-\frac{2}{x} + 4 \geq 0$$

and exclude values where $$x=0. $$Step 4: Write the expression for (g∘ƒ)(x) by substituting ƒ(x) into g(x):

$$(g∘ƒ)(x) = g(ƒ(x)) = -\frac{2}{ƒ(x)} = -\frac{2}{\sqrt{x+4}}$$ Step 5: Determine the domain of (g∘ƒ)(x). The domain includes all x-values where ƒ(x) is defined (inside the square root must be non-negative, so $$x+4 \geq 0) $$and where the denominator in g(ƒ(x)) is not zero (so $$\sqrt{x+4} 
eq 0). $$Solve these conditions to find the domain.

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7.
$ƒ (x) = \sqrt (x + 4), g (x) = - (\frac{2}{x})$

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Key Concepts

Function Composition

Domain of a Function

Square Root and Rational Function Restrictions

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7.ƒ(x)=√(x+4),g(x)=−(2x)ƒ(x)=\(\surd\)(x+4),g(x)=-(\(\frac{2}{x}\))