f as a function of x is equal to the square root of quantity 5 x plus 7, g as a function of x is equal to the square root of quantity 5 x minus 7
Find (f + g)(x)
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Ok - do you know what (f+g)(x) means? It is f(x)+g(x)
Now just plug in your function definitions, and simplify the result.
Just to make things easier, lose all the words. You have
f(x) = √(5x+7)
g(x) = √(5x-7)
To find (f + g)(x), we need to add the functions f(x) and g(x) together.
Given that f(x) = √(5x + 7) and g(x) = √(5x - 7), we can find (f + g)(x) by adding the two functions:
(f + g)(x) = f(x) + g(x)
= √(5x + 7) + √(5x - 7)
Notice that we cannot simply combine the square roots because the terms under the square roots are different.
To simplify the expression, we can rationalize the denominator. This means eliminating any square roots from the denominator. We can do this by multiplying the expression by the conjugate.
To rationalize the denominator of the second term, we multiply it by its conjugate, which is √(5x + 7):
(f + g)(x) = (√(5x + 7) + √(5x - 7)) * (√(5x + 7))/(√(5x + 7))
= (√(5x + 7)*(√(5x + 7)) + √(5x - 7)*(√(5x + 7)))/(√(5x + 7))
Simplifying further:
(f + g)(x) = (5x + 7 + √((5x + 7)*(5x - 7)))/(√(5x + 7))
= (5x + 7 + √(25x^2 - 49))/(√(5x + 7))
Therefore, (f + g)(x) = (5x + 7 + √(25x^2 - 49))/(√(5x + 7))