Let lower f left-parenthesis x right-parenthesis equals Start Root x minus 2 End Root and lower g left-parenthesis x right-parenthesis equals Start Root x plus 7 End Root. Find left-parenthesis f dot g right-parenthesis left-parenthesis x right-parenthesis. Assume all appropriate restrictions to the domain.
(1 point)
Responses
left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals x squared plus 5 x minus 14
Image with alt text: left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals x squared plus 5 x minus 14
left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals x squared plus 9 x minus 14
Image with alt text: left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals x squared plus 9 x minus 14
left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Root x squared plus 9 x minus 14 End Root
Image with alt text: left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Root x squared plus 9 x minus 14 End Root
left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Root x squared plus 5 x minus 14 End Root
Image with alt text: left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Root x squared plus 5 x minus 14 End Root
To find the composition of f and g, we substitute the function g(x) into the function f(x).
f(g(x)) = f(√(x+7)) = √(x+7) - 2
Therefore, the correct response is:
(f∘g)(x) = √(x+7) - 2
To find the product of f(x) and g(x), we need to multiply the two functions together.
f(x) = √(x - 2)
g(x) = √(x + 7)
To find the product, we can multiply the square roots:
f(x) * g(x) = (√(x - 2)) * (√(x + 7))
To simplify this expression, we can use the property √a * √b = √(a * b):
f(x) * g(x) = √((x - 2) * (x + 7))
Expanding the expression:
f(x) * g(x) = √(x^2 + 5x - 14)
Therefore, the product of f(x) and g(x) is given by:
f(x) * g(x) = √(x^2 + 5x - 14)
So, the correct answer is:
"left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Root x squared plus 5 x minus 14 End Root"
Image with alt text: left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Root x squared plus 5 x minus 14 End Root