Domain Math Example 3

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Example 3

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Find the domain of h(x)=x+1xโˆ’5h(x) = \frac{\sqrt{x + 1}}{x - 5}.

Solution

  1. 1
    The square root requires x+1โ‰ฅ0x + 1 \geq 0, so xโ‰ฅโˆ’1x \geq -1.
  2. 2
    The denominator requires xโ‰ 5x \neq 5.
  3. 3
    Combine: xโ‰ฅโˆ’1x \geq -1 and xโ‰ 5x \neq 5, giving [โˆ’1,5)โˆช(5,โˆž)[-1, 5) \cup (5, \infty).

Answer

[โˆ’1,5)โˆช(5,โˆž)[-1, 5) \cup (5, \infty)
When a function involves both a square root and a fraction, apply both constraints: the radicand must be non-negative and the denominator must be nonzero.

About Domain

The domain of a function is the complete set of allowable input values for which the function produces a defined, valid output.

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