Math  /  Algebra

QuestionLet f(x)=30xf(x)=\sqrt{30-x} and g(x)=x2xg(x)=x^{2}-x. Then the domain of fgf \circ g is equal to

Studdy Solution
To find the domain of fg(x) f \circ g(x) , ensure the expression inside the square root is non-negative:
30x2+x0 30 - x^2 + x \geq 0
Rearrange the inequality:
x2+x+300 -x^2 + x + 30 \geq 0
This can be rewritten as:
x2x300 x^2 - x - 30 \leq 0
Factor the quadratic:
(x6)(x+5)0 (x - 6)(x + 5) \leq 0
Determine the intervals where the inequality holds by testing intervals around the roots x=6 x = 6 and x=5 x = -5 .
The solution to the inequality is:
5x6 -5 \leq x \leq 6
Thus, the domain of fg(x) f \circ g(x) is:
[5,6] \boxed{[-5, 6]}

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