Written below (green dotted curve) is a graph of the function f(x)=x−2f\left(x\right)=\sqrt{x-2}f(x)=x−2. If g(x) (blue solid curve) is a reflection of f(x) about the y-axis what is the equation for g(x)?

g ( x ) = − x − 2 g\left(x\right)=\sqrt{-x-2} g ( x ) = − x − 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

Written below (green dotted curve) is a graph of the function f ( x ) = x − 2 f\left(x\right)=\sqrt{x-2} f ( x ) = x − 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> . If g(x) (blue solid curve) is a reflection of f(x) about the y-axis what is the equation for g(x) ?

Okay. Let's give this problem a try. So here we have that written below is the function f of x is equal to the square root of x minus 2. We also have g of x, which is a function graphed on here as well, and we're asked if g of x is a reflection of f of x about the y axis, what is the equation for g of x? Now whenever you see that something is reflected about the y axis, this is the same thing as saying it's reflected over the y axis. So in the same sense that we've been learning about reflections, it's like we took this entire graph, increased it at the y, and we folded it over. Now this was our original function f of x, which is this green dotted line, and we're asked what the equation is going to be for our reflected function g of x. Now to figure this out, recall that we have a situation where we're we reflect over the y, and when this happens, your original function f of x is going to become f of negative x, and the reason for this is because it's the signs of the x values that change. You'll notice if you look at one x value here we start at positive 3 right there, and then we finish at negative 3. It's the same y value that we have at on the y axis, but it changes to negative the negative x values, so that is why we say it's the inside of the function that changes. We discussed this in the video on reflections. So let's see what this new equation is going to look like. If our original function is that f of x is equal to the square root of x minus 2, like we see up here, then our transform function g of x is going to be the negative version of this. So it's gonna be f of negative x, and f of negative x is just going to be a negative sign on this x here, so it'll be negative x minus 2 or the square root of negative x minus 2, I should say. So this right here would be our transformed function, g of x, and that is how you can solve these types of problems. Hope you found this helpful. Let me know if you have any questions.

Written below (green dotted curve) is a graph of the function f ( x ) = x − 2 f\left(x\right)=\sqrt{x-2} f ( x ) = x − 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> . If g(x) (blue solid curve) is a reflection of f(x) about the y-axis what is the equation for g(x) ?

g ( x ) = − x − 2 g\left(x\right)=\sqrt{-x-2} g ( x ) = − x − 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

g ( x ) = − x − 2 g\left(x\right)=\sqrt{-x}-2 g ( x ) = − x <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> − 2

g ( x ) = x − 2 g\left(x\right)=\sqrt{x-2} g ( x ) = x − 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

g ( x ) = x − 2 g\left(x\right)=\sqrt{x}-2 g ( x ) = x <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> − 2

3. Functions

Transformations

College Algebra

3. Functions

Transformations

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Multiple Choice

Written below (green dotted curve) is a graph of the function $f\left(x\right)=\sqrt{x-2}$ . If g(x) (blue solid curve) is a reflection of f(x) about the y-axis what is the equation for g(x)?

$g\left(x\right)=\sqrt{-x-2}$

$g\left(x\right)=\sqrt{-x}-2$

$g\left(x\right)=\sqrt{x-2}$

$g\left(x\right)=\sqrt{x}-2$

Verified step by step guidance

The given function is f(x) = \sqrt{x - 2}. This function is defined for x >= 2, and its graph is the green dotted curve.

To find the reflection of f(x) about the y-axis, we need to replace x with -x in the function. This gives us g(x) = \sqrt{-x - 2}.

The domain of g(x) = \sqrt{-x - 2} is determined by the expression inside the square root being non-negative. So, -x - 2 >= 0, which simplifies to x <= -2.

The graph of g(x) is the blue solid curve, which is a reflection of the green dotted curve across the y-axis.

Therefore, the equation for g(x) is g(x) = \sqrt{-x - 2}, which matches the blue solid curve on the graph.

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