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

1 Answer
Sridhar V.
May 13, 2018


**Domain: ** [−∞<x<∞]

Using **Interval Notation: ** (−∞,∞)

**Range: ** [f(x)≥8]

Using **Interval Notation: ** [8,∞)

Explanation:


**Given: ** y=f(x)=(x+5)2+8

The Vertex Form of a quadratic function is:

y=f(x)=a(x−h)2+k, where

(h,k) is the Vertex.

**Note: ** a=1,h=−5,k=8

Step 1:

Vertex is at : (−5,8)

Note :

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

Since, a=1, the **Vertex is ** Minimum at (−5,8)

Range :f(x)≥8

Using Interval Notation : [8,∞)

Step 2:

Find Domain :

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

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

The function does not have any domain constraints.

Therefore domain is given by

−∞<x<∞

Using Interval Notation :

(−∞,∞)

Hence, the required solutions are:
**Domain: ** [−∞<x<∞]

Using **Interval Notation: ** (−∞,∞)

**Range: ** [f(x)≥8]

Using **Interval Notation: ** [8,∞)

Step 3:

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

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Hope it helps.

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