11-1 Square-Root Functions
Preview
Warm Up
California Standards
Lesson Presentation
11-1 Square-Root Functions
Warm Up
Find each square root.
1.
2.
3.
4.
undefined
12
–20
6
Solve each inequality.
x≥2
x ≥ –5
5. x + 5 ≥ 0
6. 0 ≤ 4x – 8
7. 0 ≤ 3x x ≥ 0
8. 10 – 3x ≥ 0
Compare. Write <, >, or =.
9.
< 7
10.
>3
11-1 Square-Root Functions
California
Standards
Extension of
2.0 Students understand
and use such operations as taking the opposite,
finding the reciprocal, taking a root, and raising
to a fractional power. They understand and use the
rules of exponents.
Also covered: 17.0
11-1 Square-Root Functions
Vocabulary
square-root function
11-1 Square-Root Functions
The function
gives the speed in feet per
second of an object in free fall after falling x feet.
This function is different from others you have
seen so far. It contains a variable under the
square-root sign,
.
11-1 Square-Root Functions
11-1 Square-Root Functions
Additional Example 1A: Evaluating Square-Root
Functions
The function
gives the speed in feet per
second of an object in free fall after falling x
feet.
Find the speed of an object in free fall after it
has fallen 16 feet.
Write the speed function.
Substitute 16 for x.
= 8(4)
Simplify.
= 32
After an object has fallen 16 feet, its speed is 32 ft/s.
11-1 Square-Root Functions
Additional Example 1B: Evaluating Square-Root
Functions
The function
gives the speed in feet per
second of an object in free fall after falling x
feet.
Find the speed of an object in free fall after it
has fallen 20 feet. Round your answer to the
nearest tenth.
Write the speed function.
Substitute 20 for x. Use a
calculator to find the square
root.
Simplify.
≈ 35.8
After an object has fallen 20 feet, its speed is about
35.8 ft/s.
11-1 Square-Root Functions
Check It Out! Example 1a
Find the speed of an object in free fall after it
has fallen 25 feet.
Write the speed function.
Substitute 25 for x.
= 8(5)
Simplify.
= 40
After an object has fallen 25 feet, its speed is 40 ft/s.
11-1 Square-Root Functions
Check It Out! Example 1b
Find the speed of an object in free fall after it has
fallen 15 feet. Round your answer to the nearest
hundredth.
Write the speed function.
Substitute 15 for x. Use a
calculator to find the square
root.
≈ 30.98
Simplify.
After an object has fallen 15 feet, its speed is about
30.98 ft/s.
11-1 Square-Root Functions
Recall that the square root of a negative number
is not a real number. The domain (x-values) of a
square-root function is restricted to numbers that
make the value under the radical sign greater
than or equal to 0.
11-1 Square-Root Functions
Additional Example 2: Finding Domain of SquareRoot Functions
Find the domain of each square-root function.
A.
x– 4 ≥
0
x
4
≥
B.
x +3 ≥
x
0
≥ –3
The expression under the radical
sign must be greater than or
equal to 0.
Solve the inequality. Add 4 to both
sides.
The expression under the radical
sign must be greater than or
equal to 0.
Solve the inequality. Subtract 3
from both sides.
11-1 Square-Root Functions
Check It Out! Example 2
Find the domain of
2x – 1
≥
0
2x
≥
1
.
The expression under the radical
sign must be greater than or
equal to 0.
Solve the inequality. Add 1 to both
sides.
Divide both sides by 2.
11-1 Square-Root Functions
The function for
is graphed at right. Notice
there are no x-values to the
left of 0 because the domain
is x ≥ 0.
Remember that the symbol
indicates the
positive square root only. For this reason the range
is y ≥ 0.
of
11-1 Square-Root Functions
Additional Example 3: Graphing Square-Root
Functions
Graph
.
Step 1 Find the domain of the function.
x≥3
The expression under the radical sign
must be greater than or equal to 3.
11-1 Square-Root Functions
Additional Example 3 Continued
Graph
.
Step 2 Choose x-values
greater than or equal to
3 and generate ordered
pairs.
Step 3 Plot the points.
Then connect them with
a smooth curve.
x
3
0
4
1
5
1.41
7
2



11-1 Square-Root Functions
Helpful Hint
In Example 3, when generating ordered pairs,
choose x-values that make the expression under
the radical sign a perfect square.
11-1 Square-Root Functions
Check It Out! Example 3
Graph square root function.
Step 1 Find the domain of the function.
x≥4
The expression under the radical sign
must be greater than or equal to 4.
11-1 Square-Root Functions
Check It Out! Example 3 Continued
Graph
Step 2 Choose x-values
greater than or equal to
4 and generate ordered
pairs.
Step 3 Plot the points.
Then connect them with
a smooth curve.
x
4
0
5
1
6
1.41
8
2



11-1 Square-Root Functions
Lesson Quiz: Part I
1. Use the formula
to find the radius of a circle
whose area is 28 in2. Use 3.14 for . Round your
answer to the nearest tenth of an inch. 3.0 in.
Find the domain of each square-root function.
2.
x≥0
3.
4.
x≥5
11-1 Square-Root Functions
Lesson Quiz: Part II
Graph each square-root function.
5.
6.