f(x) = squrt 5 −x g(x) = squrt x^2 − 1 What are the domains of f+g, f-g, fxg, f/g in interval notation? Please explain?
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F(x) = squrt 5 −x, g(x) = squrt x^2 − 1. What are the domains of f+g, f-g, fxg, f/g in interval notation? Please explain.
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For both f + g and f - g, 5 - x is non-negative for all x <= 5. x^2 - 1 is non-negative for all x <= -1 or x >= 1.
Total interval domain: (-infinity, -1] U [1, 5].
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For f × g, you can combine the two radicands as one product; i.e., sqrt((5 - x)(x^2 - 1)). -1, 1, and 5 are the key values of x that are the boundaries and the intervals you see must ensure a non-negative product. Testing a value in each interval, you'd find that all x <= -1 works, -1 < x < 1 fails, 1 <= x <= 5 works, and x > 5 fails.
Total interval domain is the same as the first case: (-infinity, 1] U [1, 5].
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For f / g, this is simply constructing a fraction with 5 - x as the numerator and x^2 - 1 as the denominator inside their respective square roots.
sqrt(5 - x) still keeps its domain of x <= 5. But not sqrt(x^2 - 1); while x < -1 and x > 1 stay as part of the domain, 1 and -1 themselves are EXCLUDED because the denominator would be 0.
Total interval domain: (-infinity, -1) U (1, 5].
Do you mean √5-x, or √(5-x) ?
√x²-1, or √(x²-1) ?