Question

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. ƒ(x)=1/(x+4), g(x)=-(1/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: Find (ƒ∘g)(x) by substituting g(x) into ƒ. Since ƒ(x) = \frac{1}{x+4} and g(x) = -\frac{1}{x}, replace x in ƒ(x) with g(x) to get ƒ(g(x)) = \frac{1}{\left(-\frac{1}{x}ight) + 4}. Step 3: Determine the domain of (ƒ∘g)(x). Start with the domain of g(x), which excludes x = 0 because of division by zero. Then, ensure the expression inside ƒ, which is g(x), does not make the denominator zero in ƒ(g(x)). So solve \left(-\frac{1}{x}ight) + 4 
eq 0 to find any additional restrictions. Step 4: Find (g∘ƒ)(x) by substituting ƒ(x) into g. Since g(x) = -\frac{1}{x} and ƒ(x) = \frac{1}{x+4}, replace x in g(x) with ƒ(x) to get g(ƒ(x)) = -\frac{1}{\frac{1}{x+4}}. Step 5: Determine the domain of (g∘ƒ)(x). Start with the domain of ƒ(x), which excludes x = -4 due to division by zero. Then ensure the input to g, which is ƒ(x), does not cause division by zero in g(ƒ(x)). So find values of x for which \frac{1}{x+4} 
eq 0 and also check for any other restrictions.

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. ƒ(x)=1/(x+4), g(x)=-(1/x)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Function Composition

Domain of a Function

Rational Functions and Restrictions