Solve.
1. x5/2 = 32
2. x2/3 + 15 = 24
x2/3 = 9
(x5/2 )2/5 = 322/5
(x2/3)3/2 = 93/2
x = (321/5)2
x = 22
x=4
3. 4x3/4 = 108
x3/4 = 27
(x3/4)4/3 = 274/3
x = (271/3)4
x = 34
x = 81
x = (91/2)3
x = 33
x = 27
7.5 Solving Radical Equations
Objective - To be able to solve square root
and other radical equations.
State Standard 15.0 - Students will be able to
solve radical expressions
7.5 Solving Radical Equations
Example 1:
Solve:
3
x–4=0
3
(
3
x =4
3
x) = 4
3
x = 64
Example 2:
Solve: 2x3/2 = 250
x3/2 = 125
(x3/2)2/3 = 1252/3
x = 125 2/3
x = (1251/3)2
x = 52
x = 25
Example 3:
Solve:
3x + 2 – 2 x = 0
3x + 2 = 2 x
( 3x+2)2 = ( 2
3x + 2 = 4x
2 = x
x )2
Example 4:
Solve: x – 4 = 2x
(x – 4)2 = ( 2x )2
x2 – 8x + 16 = 2x
x2 – 10x + 16 = 0
(x – 2)(x – 8) = 0
x – 2 = 0 and x – 8 = 0
x = 2 and x = 8
7.6 Warm-Up
Perform indicated operations
2
1.(2x+7)
2
4x +28x+49
2. 2
4
2
7.6 Function Operations
Operations on Functions
Operation
Definition
Example f(x)=2x, g(x)=x+1
Addition
f(x)+g(x)
2x+(x+1) = 3x+1
Subtraction
f(x) – g(x)
2x – (x+1) = x – 1
Multiplication (f(x))(g(x))
Division
f(x)
g(x)
2x(x+1) = 2x2 + 2x
2x
(x+1)
Example 1
Adding and Subtracting
Functions
f(x)=3x, g(x)=x+2
f(x)+g(x)
f(x) - g(x)
3x+(x+2)
3x - (x+2)
4x+2
2x - 2
Example 2
Multiplication and Division Functions
f(x)=3x, g(x)=x+2
(f(x))(g(x))
f(x)/g(x)
(3x)(x+2)
2
3x +6x
3x/(x+2)
f(x) = x + 4, g(x) = 3x
h(x) = 2(f(x)) + 2(g(x))
h(x) = 4(f(x)) - 3(g(x))
h(x) = (2x+8) + (6x)
h(x) = (4x+16) - (9x)
h(x) = 8x + 8
h(x) = -5x + 16
f(x)=x-1, g(x)= 2x
h(x)=3(f(x)) / 2g(x)
h(x)=(-2(f(x))(g(x))
h(x)=3(x-1) / 2(2x)
h(x)=(-2(x-1))(2x)
h(x)=3x-3 / 4x
h(x)=(-2x+2)(2x)
2
-4x + 4x
Composition of two functions
The composition of the function f with the function g
is:
f(g(x)) or (f ο g)(x)
This is read as:
f of g of x
Example 3a
f(x) = 2x and g(x) = 3x + 1
f(g(x))
f(3x + 1)
g(f(x))
g(2x)
2(3x + 1)
3(2x) + 1
6x + 2
6x + 1
Example 3b
-1
f(x) = 3x and g(x) = 2x – 1
f(g(x))
f(2x-1)
3(2x-1)
3
2x-1
-1
Example 3c
3
f(x) = x and g(x) = x2 + 7
(g ο f)(2)
g(f(x))
g(x3)
g(23)
g(8)
82 + 7
64 + 7
71
Evaluate the compositions if:
f(x) = x + 2 g(x) = 3
h(x) = x2 + 3
1. f(g(x))
2. h(f(x))
f(3)
f(x) = x + 2
3+2
h(x + 2)
h(x) = x2 + 3
(x + 2)2 + 3
x2 + 4x + 4 + 3
5
x2 + 4x + 7
3. h(f(g(x)))
h(f(3))
h(3 + 2)
h(5)
52 + 3
25 + 3 = 28
Goal - Find inverses of linear functions.
State Standard 24.0 – Students solve problems
involving inverse functions
Solving for the Inverse
STEP 1
Switch the “y” and the “x” values.
STEP 2
Solve for “y”.
Example 1:
Find the inverse of 10y +2x = 4
10x + 2y = 4
2y = -10x + 4
2
2
y = -5x + 2
Answer: y -1 = -5x + 2
Example 2:
Find the inverse of y = -3x + 6
x = -3y + 6
–6
–6
x – 6 = -3y
–3
–3
y = (-1/3)x + 2
Answer:
y -1 = (-1/3)x + 2
Example 3:
Find the inverse of the function: f(x) = x5
y=x5
x=y5
5
f -1(x)=
x = y
5
x
1) Identify the
domain and range
Input Output
2
3
5
-1
-2
6
Domain = -1, 2, 5 & 6
Range = -2 & 3
2) Graph
y = -2x2 + 3
Objective- Students will learn to graph functions of the form
y = a x – h + k and y = a 3 x – h + k.
y
y
(1,1)
(0,0)
Domain: x > 0, Range: y > 0
(1,1)
x
(-1,-1)
(0,0)
x
Domain and range: all real numbers
 Example 1
Comparing Two Graphs
Describe how to create the graph of y =
from the graph of y = x .
x+2 –4
Solution
h = -2 and k = -4
shift the graph to the left 2 units &
down 4 units
Graphs of Radical
Functions
To graph y = a x - h + k or y = a 3 x - h + k,
follow these steps.
STEP u Sketch the graph of y = a
x or y = a
3
x.
STEP v Shift the graph h units horizontally and k units vertically

Example 2
Graphing a Square Root
Graph y = -3
x–1+3.
(1, 3)
(0,0)
(2,0)
(1,-3)
Solution
1) Sketch the graph of
y = -3 x (dashed).
It begins at the origin
and passes through
point (1,-3).
2) For y = -3 x – 1 + 3,
h = 1 & k = 3.
Shift both points 1 to
the right and 3 up.

On White Board
Graphing a Square Root
Graph y = 2
x–2+1.
(3,3)
(1,2)
(2, 1)
(0,0)
 Example 3
Graphing a Cube Root
Graph y = 2
3
x+3–4.
(1, 2)
(0,0)
(-1,-2)
(-2,-2)
(-3,-4)
(-4,-6)
Solution
1) Sketch the graph of
y = 2 3 x (dashed).
It passed through the
origin & the points
(1, 2) & (-1, -2).
2) For y = 2 x + 3 – 4,
h = -3 & k = -4.
Shift the three
points Left 3 and
Down 4.
 On White Board
Graphing a Cube Root
Graph y = 3 3 x – 2 + 1
(1, 3)(3,4)
(0,0)
(-1,-3)
(2,1)
(1,-2)