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# Domain

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What is the smallest integer value of c such that the function f(x) = (x^2 + 1)/(x^2 - 4x + c)  has a domain of all real numbers?

Jul 13, 2021

#1
+208
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the only way the expression $$\frac{x^2\:+\:1}{x^2\:-\:4x\:+\:c}$$ is not real is if the denominator equals zero

$$x^2\:-\:4x\:+\:c=0$$

discriminant is:

$$(-4)^2-4c \\=16-4c$$

to have no real solutions:

$$16-4c<0 \\4c>16 \\c>4$$

smallest integer value is c=5

JP

Jul 14, 2021

#1
+208
+1

the only way the expression $$\frac{x^2\:+\:1}{x^2\:-\:4x\:+\:c}$$ is not real is if the denominator equals zero

$$x^2\:-\:4x\:+\:c=0$$

discriminant is:

$$(-4)^2-4c \\=16-4c$$

to have no real solutions:

$$16-4c<0 \\4c>16 \\c>4$$

smallest integer value is c=5

JP

JKP1234567890 Jul 14, 2021