Questions: Determine whether (y^2=2 x^2-1) is a function. Select the correct answer below: yes no

Transcript text: Determine whether $y^{2}=2 x^{2}-1$ is a function. Select the correct answer below: yes no

Solution

Solution Steps

To determine if $ y^2 = 2x^2 - 1 $ is a function, we need to check if for every value of $ x $, there is exactly one corresponding value of $ y $. If there are multiple values of $ y $ for a single $ x $, then it is not a function.

Solution Approach

Rearrange the equation to solve for $ y $.
Check if the resulting expression for $ y $ gives a unique value for each $ x $.

Step 1: Rearrange the Equation

We start with the equation: \[ y^2 = 2x^2 - 1 \]

Step 2: Solve for $ y $

To determine if this relation is a function, we solve for $ y $: \[ y = \pm \sqrt{2x^2 - 1} \]

Step 3: Analyze the Solutions

The solutions are: \[ y = \sqrt{2x^2 - 1} \] \[ y = -\sqrt{2x^2 - 1} \]

For each value of $ x $, there are two possible values of $ y $ (one positive and one negative), except when $ 2x^2 - 1 = 0 $. This means that for most values of $ x $, there are two corresponding values of $ y $.