Graph f(x) = square root of 1-x

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

Step 1

Find the domain for $y = \sqrt{1 - x}$ 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 $\sqrt{1 - x}$ greater than or equal to $0$ to find where the expression is defined.

$1 - x \geq 0$

Solve for $x$ .

Subtract $1$ from both sides of the inequality.

$- x \geq - 1$

Divide each term in $- x \geq - 1$ by $- 1$ and simplify.

Divide each term in $- x \geq - 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 1}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{- 1}{- 1}$

Simplify the right side.

Divide $- 1$ by $- 1$ .

$x \leq 1$

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

Interval Notation:

$(- \infty, 1]$

Set-Builder Notation:

${x | x \leq 1}$

Interval Notation:

$(- \infty, 1]$

Set-Builder Notation:

${x | x \leq 1}$

Step 2

To find the radical expression end point, substitute the $x$ value $1$ , which is the least value in the domain, into $f (x) = \sqrt{1 - x}$ .

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

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

Simplify the result.

Multiply $- 1$ by $1$ .

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

Subtract $1$ from $1$ .

$f (1) = \sqrt{0}$

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

$f (1) = \sqrt{0^{2}}$

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

$f (1) = 0$

The final answer is $0$ .

$0$

Step 3

The radical expression end point is $(1, 0)$ .

$(1, 0)$

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 $f (x) = \sqrt{1 - x}$ . In this case, the point is $(- 1, \sqrt{2})$ .

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

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

Simplify the result.

Multiply $- 1$ by $- 1$ .

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

Add $1$ and $1$ .

$f (- 1) = \sqrt{2}$

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

$y = \sqrt{2}$

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

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

$f (0) = \sqrt{1 - (0)}$

Simplify the result.

Multiply $- 1$ by $0$ .

$f (0) = \sqrt{1 + 0}$

Add $1$ and $0$ .

$f (0) = \sqrt{1}$

Any root of $1$ is $1$ .

$f (0) = 1$

The final answer is $1$ .

$y = 1$

The square root can be graphed using the points around the vertex $(1, 0), (- 1, 1.41), (0, 1)$

$\begin{array}{c} x & y \\ - 1 & 1.41 \\ 0 & 1 \\ 1 & 0 \end{array}$

Step 5

Graph f(x) = square root of 1-x

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