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Math  /  Algebra

QuestionWhat is the domain of each function? a) y=2x−5y=2 x-5 b) y=x2+5y=x^{2}+5 c) y=3x+5y=\frac{3}{x+5} d) y=7x2−4y=\frac{7}{x^{2}-4}

Studdy Solution

STEP 1

1. The domain of a function is the set of all possible input values (x-values) for which the function is defined.
2. For polynomial functions, the domain is all real numbers.
3. For rational functions, the domain excludes values that make the denominator zero.

STEP 2

1. Determine the domain of the linear function y=2x−5 y = 2x - 5 .
2. Determine the domain of the quadratic function y=x2+5 y = x^2 + 5 .
3. Determine the domain of the rational function y=3x+5 y = \frac{3}{x+5} .
4. Determine the domain of the rational function y=7x2−4 y = \frac{7}{x^2 - 4} .

STEP 3

For the linear function y=2x−5 y = 2x - 5 :
- Linear functions are defined for all real numbers because there are no restrictions like division by zero or square roots of negative numbers.
Domain: (−∞,∞) (-\infty, \infty)

STEP 4

For the quadratic function y=x2+5 y = x^2 + 5 :
- Quadratic functions are also defined for all real numbers as they do not involve division by zero or square roots.
Domain: (−∞,∞) (-\infty, \infty)

STEP 5

For the rational function y=3x+5 y = \frac{3}{x+5} :
- The function is undefined where the denominator is zero. Solve for x+5=0 x+5 = 0 .
x+5=0 x + 5 = 0 x=−5 x = -5
- The domain excludes x=−5 x = -5 .
Domain: (−∞,−5)∪(−5,∞) (-\infty, -5) \cup (-5, \infty)

STEP 6

For the rational function y=7x2−4 y = \frac{7}{x^2 - 4} :
- The function is undefined where the denominator is zero. Solve for x2−4=0 x^2 - 4 = 0 .
x2−4=0 x^2 - 4 = 0 (x−2)(x+2)=0 (x - 2)(x + 2) = 0 x=2orx=−2 x = 2 \quad \text{or} \quad x = -2
- The domain excludes x=2 x = 2 and x=−2 x = -2 .
Domain: (−∞,−2)∪(−2,2)∪(2,∞) (-\infty, -2) \cup (-2, 2) \cup (2, \infty)
The domains are: a) (−∞,∞) (-\infty, \infty) b) (−∞,∞) (-\infty, \infty) c) (−∞,−5)∪(−5,∞) (-\infty, -5) \cup (-5, \infty) d) (−∞,−2)∪(−2,2)∪(2,∞) (-\infty, -2) \cup (-2, 2) \cup (2, \infty)

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