How do you find the domain and range of f(x) = (x + 5)^2 + 8f(x)=(x+5)2+8?

1 Answer
May 13, 2018

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Domain: color(blue)([-oo < x < oo ][<x<]

Using Interval Notation: color(blue)((-oo,oo)(,)

Range: color(blue)([f(x)>=8 ][f(x)8]

Using Interval Notation: color(blue)([8,oo)[8,)

Explanation:

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Given: color(red)(y=f(x)=(x+5)^2+8y=f(x)=(x+5)2+8

The Vertex Form of a quadratic function is:

color(green)(y=f(x)=a(x-h)^2+ky=f(x)=a(xh)2+k, where

(h,k)(h,k) is the Vertex.

Note: color(blue)(a=1, h=-5, k=8a=1,h=5,k=8

color(green)("Step 1: "Step 1:

Vertex is at : color(blue)((-5,8)(5,8)

Note :

If a>0a>0, then the Vertex is a Minimum Value.

Since, a=1a=1, the Vertex is color(blue)("Minimum at "(-5,8)Minimum at (5,8)

"Range :"f(x)>= 8Range :f(x)8

Using Interval Notation : [8, oo)[8,)

color(green)("Step 2: "Step 2:

Find Domain :

Domain of f(x)f(x) is the set of all input values for which the given function is real and well-defined.

color(blue)(f(x) = (x+5)^2+8f(x)=(x+5)2+8 does not have any undefined points.

The function does not have any domain constraints.

Therefore domain is given by

color(blue)(-oo < x < oo<x<

Using Interval Notation :

color(blue)((-oo , oo)(,)

Hence, the required solutions are:
Domain: color(blue)([-oo < x < oo ][<x<]

Using Interval Notation: color(blue)((-oo,oo)(,)

Range: color(blue)([f(x)>=8 ][f(x)8]

Using Interval Notation: color(blue)([8,oo)[8,)

color(green)("Step 3: "Step 3:

Draw the graph of color(red)(y=f(x)=(x+5)^2+8y=f(x)=(x+5)2+8 to verify the solutions:

enter image source here

Hope it helps.