Name: Algebraic Operations: Functions
Uploaded: 2019-07-19
Duration: 7 min 10 s
Description: Now that we know what functions are, what can we do with them? Can we do operations with them? Yes! What won't these functions do? Let us learn more here.

Question

Find domain of

f (x) = \frac{\sqrt{\cos x - \frac{1}{2}}}{\sqrt{6 + 35 x - 6 x^{2}}}

.

Accepted Answer

Given: $f (x) = \frac{\sqrt{\cos x - \frac{1}{2}}}{\sqrt{6 + 35 x - 6 x^{2}}}$ .

Let $g (x) = \sqrt{\cos x - \frac{1}{2}}$ , domain set $D_{1}$ and $h (x) = \sqrt{6 + 35 x - 6 x^{2}}$ , domain set $D_{2}$ .

$C a s e - 1$ : Finding the domain of $g (x) \to D_{1}$ .

$\cos x - \frac{1}{2} \geq 0$ .

$\cos x \geq \frac{1}{2} . . . (i)$ , the solution set of the $(i)$ , is the domain of $g (x)$

To solve $(i)$ , draw the graph of $\cos x$ and $y = \frac{1}{2}$ .

From the graph we can observe that between $- \frac{π}{3} \leq x \leq \frac{π}{3}$ , $\cos x \geq \frac{1}{2}$ , so

$D_{1} \to 2 n π - \frac{π}{3} \leq x \leq 2 nπ + \frac{π}{3}, n \in I . . . (i i)$ (for generalizing the solution adding $2 n π$ )

$C a s e - 2$ : Finding the domain of $h (x) \to D_{2}$ .

$6 + 35 x - 6 x^{2} > 0$

To solve the $(i i i)$ , we draw the graph of $y = 6 + 35 x - 6 x^{2}$

Since coefficients of $x^{2} < 0$ , so the graph is downward parabola.

From the graph, the domain $D_{2} \to - \frac{1}{6} < x < 6 . . . (i i i)$

Now the domain of the function $f (x)$ , is given by $D = D_{1} \cap D_{2}$ , so from $(i i)$ and $(i i i)$

The domain of the function $f (x)$ , is $(- \frac{1}{6}, \frac{π}{3}] \cup [\frac{5 π}{3}, 6)$ .

Find domain of $f (x) = \frac{\sqrt{\cos x - \frac{1}{2}}}{\sqrt{6 + 35 x - 6 x^{2}}}$ .

Important Questions on Functions

Learn and Prepare for any exam you want !

Find domain of fx=cosx-126+35x-6x2.

Important Questions on Functions

Learn and Prepare for any exam you want !

Find domain of $f (x) = \frac{\sqrt{\cos x - \frac{1}{2}}}{\sqrt{6 + 35 x - 6 x^{2}}}$ .