Find Range Algebraically

[nycmath]

I need a step by step explanation.

Find the range algebraically.

g(x) = sqrt{4 - x^2}

[Eigenvalue]

Rewrite the equation as y=[tex]\sqrt{4-x²}[/tex]

Solve for x as a value of y

y²=4-x²
x²=4-y²
x=[tex]\sqrt{4-y²}[/tex]

4-y²[tex]\ge[/tex]0

y²[tex]\le[/tex]4

-2[tex]\le[/tex]x[tex]\le[/tex]2

As y is a square root, y[tex]\ge[/tex]0

[0,2]

[nycmath]

Rewrite the equation as y=[tex]\sqrt{4-x²}[/tex]

Solve for x as a value of y

y²=4-x²
x²=4-y²
x=[tex]\sqrt{4-y²}[/tex]

4-y²[tex]\ge[/tex]0

y²[tex]\le[/tex]4

-2[tex]\le[/tex]x[tex]\le[/tex]2

As y is a square root, y[tex]\ge[/tex]0

[0,2]

You lost me after x = sqrt{4 - y^2}.

Why did x disappear and then reappeared -2 <= x <= 2?

[Eigenvalue]

It was a typo; it was meant to be -2[tex]\le[/tex]x[tex]\le[/tex]2

In order to find the domain, you have to rewrite x as a function of y

[nycmath]

It was a typo; it was meant to be -2[tex]\le[/tex]x[tex]\le[/tex]2

In order to find the domain, you have to rewrite x as a function of y

So, basically we solve for x in terms of y. This yields the range.