Chapter 3: Q. 24 (page 171)

What is the domain of the function $f (x) = \sqrt{x - 3 x^{2}}$ ?

Short Answer

Expert verified

The domain of the function is $0 \leq x \leq \frac{1}{3}$ .

Step by step solution

Step 1. Given Information

The given function is $f (x) = \sqrt{x - 3 x^{2}}$ .

Step 2. Find the Intercepts and Vertex

The square root function $\sqrt{x - 3 x^{2}}$ is defined when $x - 3 x^{2} \geq 0$ .
Consider $g (x) = x - 3 x^{2}$ and factor the expression.

$g (x) = x (1 - 3 x)$

$g (x) = 0$ implies $x = 0$ or $(1 - 3 x) = 0$ .
So, the x -intercepts of the curve are $(0, 0), (\frac{1}{3}, 0)$ .
The value of $g (0) = 0$ , so, the y-intercept is $(0, 0)$ .
Find the vertex of the curve as follows:

$\begin{array}{rcl} x & = & \frac{- b}{2 a} \\ = & \frac{- 1}{- 6} \\ = & \frac{1}{6} \end{array}$

So, the vertex is also at $(\frac{1}{6}, \frac{1}{12})$ .

Step 3. Plot the function

Plot the graph of the function by using the obtained intercepts and vertex.

Step 4. Find the region above x-axis

From the graph, the region on and above the horizontal axis is $0 \leq x \leq \frac{1}{3}$ .
The inequality is of the greater than or equal to type, so the solution set is $0 \leq x \leq \frac{1}{3}$ .

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What is the domain of the function $f (x) = \sqrt{x - 3 x^{2}}$ ?

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Find the Intercepts and Vertex

Step 3. Plot the function

Step 4. Find the region above x-axis

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What is the domain of the functionf(x)=x-3x2?

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Find the Intercepts and Vertex

Step 3. Plot the function

Step 4. Find the region above x-axis

One App. One Place for Learning.

Most popular questions from this chapter

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What is the domain of the function $f (x) = \sqrt{x - 3 x^{2}}$ ?