Evaluate f+g, f−g, fg and fg of the function f(x)=√9−x2 and g(x)=√x2−4 and find their domain
For f+g,
f+g=f(x)+g(x)f+g=√9−x2+√x2−4
The radicand can't be a negative value. So we factor 9−x2=(3−x)(3+x) and x2−4=(x−2)(x+2). Thus the domain of f(x)+g(x) is [−3,−2]⋃[2,3]
For f−g
f−g=f(x)−g(x)f−g=√9−x2−√x2−4
The radicand can't be a negative value. So we factor 9−x2=(3−x)(3+x) and x2−4=(x−2)(x+2). Thus the domain of f(x)−g(x) is [−3,−2]⋃[2,3]
For fg
fg=f(x)⋅g(x)fg=(√9−x2)(√x2−4)Substitute f(x)=√9−x2 and g(x)=√x2−4fg=√(9−x2)(x2−4)Apply FOIL methodfg=√9x2−36−x4+4x2Combine like termsfg=√13x2−x4−36
The radicand can't be a negative value. So we factor 9−x2=(3−x)(3+x) and x2−4=(x−2)(x+2). Thus the domain of f(x)⋅g(x) is [−3,−2]⋃[2,3]
For fg
fg=f(x)g(x)fg=√9−x2√x2−4Substitute f(x)=√9−x and g(x)=√x2−4fg=√9−x2x2−4
The function fg can't have a denominator equal to zero and the radicand can't be a negative value. So we factor 9−x2=(3−x)(3+x) and x2−4=(x−2)(x+2). Thus, the domain of fg is [−3,−2)⋃(2,3]
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