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Chapter 2: Problem 62

Find the domain of each function. $$f(x)=\frac{1}{\sqrt{4 x^{2}-9 x+2}}$$

Short Answer

Expert verified
The domain of the function is all x for which \(x鈮_{1}\) or \(x鈮_{2}\) excluding the roots that make the denominator zero. The exact values depend on the solutions to the quadratic equation \(4x^2 - 9x + 2 = 0\).

Step by step solution

01

Identify Values For Which The Function Is Undefined

The function \(f(x) = \frac{1}{\sqrt{4x^2 - 9x + 2}}\) is undefined where the denominator is zero or negative. Set the radicand (the expression under the square root) to be equal to zero to find x-values that will make the function undefined:\n\[\n4x^2 - 9x + 2 = 0\n\]
02

Solve Quadratic Equation

This is a quadratic equation, we can find its solution either by factoring, completing the square or using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The solutions provide the values of x where the numerator will become zero:\n\[\nx = \frac{9 \pm \sqrt{(9)^2 - 4*(4)*(2)}}{2*4}\n\]
03

Calculate the Discriminant

Calculate the value under the root (the discriminant) and if it is positive, find the roots of the quadratic equation. If it is negative, it implies that no real roots exist and the denominator can never be zero, therefore all x-values are possible.\n\[\nD = (9)^2 - 4*(4)*(2)\n\]
04

Find Root of the Quadratic Equation

If the discriminant is positive, find the roots. The roots of the equation, \[x_{1,2} = \frac{9 \pm \sqrt{D}}{8}\], are the x-values where the denominator will become zero. These value should be excluded from the domain of the function.
05

Identify Range Where Radicand is Non-Negative

As for the square root, recall that the value under the square root (radicand) must be greater than or equal to zero. The quadratic expression will be non-negative outside the roots, i.e., \(x鈮_{1} or x鈮_{2}\), where \(x_{1}\) is the smaller root and \(x_{2}\) is the larger root.

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