Graph y=x^4- square root of x

$y = x^{4} - \sqrt{x}$

Step 1

Find the domain for $y = x^{4} - \sqrt{x}$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

Set the radicand in $\sqrt{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 2

To find the radical expression end point, substitute the $x$ value $0$ , which is the least value in the domain, into $f (x) = x^{4} - \sqrt{x}$ .

Replace the variable $x$ with $0$ in the expression.

$f (0) = {(0)}^{4} - \sqrt{0}$

Simplify the result.

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 - \sqrt{0}$

Rewrite $0$ as $0^{2}$ .

$f (0) = 0 - \sqrt{0^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$f (0) = 0 - 0$

Multiply $- 1$ by $0$ .

$f (0) = 0 + 0$

Add $0$ and $0$ .

$f (0) = 0$

The final answer is $0$ .

$0$

Step 3

The radical expression end point is $(0, 0)$ .

$(0, 0)$

Step 4

Select a few $x$ values from the domain. It would be more useful to select the values so that they are next to the $x$ value of the radical expression end point.

Substitute the $x$ value $1$ into $f (x) = x^{4} - \sqrt{x}$ . In this case, the point is $(1, 0)$ .

Replace the variable $x$ with $1$ in the expression.

$f (1) = {(1)}^{4} - \sqrt{1}$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = 1 - \sqrt{1}$

Any root of $1$ is $1$ .

$f (1) = 1 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$f (1) = 1 - 1$

Subtract $1$ from $1$ .

$f (1) = 0$

The final answer is $0$ .

$y = 0$

Substitute the $x$ value $2$ into $f (x) = x^{4} - \sqrt{x}$ . In this case, the point is $(2, 16 - \sqrt{2})$ .

Replace the variable $x$ with $2$ in the expression.

$f (2) = {(2)}^{4} - \sqrt{2}$

Simplify the result.

Raise $2$ to the power of $4$ .

$f (2) = 16 - \sqrt{2}$

The final answer is $16 - \sqrt{2}$ .

$y = 16 - \sqrt{2}$

The square root can be graphed using the points around the vertex $(0, 0), (1, 0), (2, 14.59)$

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 0 \\ 2 & 14.59 \end{array}$

Step 5

Graph y=x^4- square root of x

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