Find all values of 𝑐 such that 𝑓(𝑥)=𝑥+9/𝑥^2+2𝑐𝑥+4 has domain 𝑅. (Express numbers in exact form. Use symbolic notation and fractions where needed. Give the inequality in terms of 𝑐.

Answer 1

I'm going to assume you wanted this problem to read (x+9)/(x^2 + 2cx+ 4) otherwise, there is no solution because x<>0 as you have it written.

For the domain to include all real numbers, the denominator cannot = 0. This means that the function can NEVER touch the x-axis. Mathematically, it means that the roots to the equation are imaginary or the discriminant from the quadratic formula is <0 (The discriminant, recall is b^2-4ac.

For the quadratic formula, the coefficients are a=1 b=2c, c=4. Therefore (2c)^2 - 4(1)(4) <0

or

4c^2 <16 or

c^2 <4.

That means that c must fall between -2 and 2 for the value of the denominator to never equal 0.