Root functions

Root functions
1
Precalculus
Root function f (x) = √x
y
C
B
A
x
Fig. 1: Root function f (x) = √x
The domain (D) and the range (R) of the root function f  x =  x
are
D  f  = [ 0, ∞ ) ,
R  f  = [ 0, ∞ ) .
Remember, in the field of the real numbers, roots are only defined for
positive radicands
A = 1, 1 ,
2
B =  4, 2 ,
C = 9, 3
Precalculus
Root functions
y
g(x)
D
C
B
f(x)
A
x
Fig. 2: Root functions f (x) and g(x)
f x =
A = 1, 1 ,
3
x
,
B =  4, 2 ,
g x = 2 x
C = 1, 2 ,
D = 4, 4
Precalculus
Root functions
y
h(x)
f(x)
g(x)
A
B
-3
C
+2
x
Fig. 3: Root functions f (x), g(x) and h(x)
f x =
x
,
g  x =
A = −2, 1 ,
x
−2 ,
B = 1, 1 ,
h  x =
x
3
C =  3, 1

BC
f x =
f x =
4
x


x


g x =
x − 2
h x =
x  3

BA
Precalculus
Root functions
y
h(x)
B
f(x)
O
g(x)
x
A
Fig. 4: Root functions f (x), g(x) and h(x)
f x =
x
,
g  x =
x
− 1,
hx =
x
2

OA
f x =
x


g x  =
x
−1
hx =
x
2

OB
f x =
5
x


Precalculus
Root functions
y
h(x)
f(x)
x
g(x)
Fig. 5: Root functions f (x), g(x) and h(x)
f x =
x
,
g  x = − x ,
h  x  = −x
x − Achse
f x =
x


g  x = −x
y − Achse
f x =
6
x


h  x  = −x
Precalculus
Root functions
y
g(x)
A
f(x)
O
x
h(x)
Fig. 6: Root functions f (x), g(x) and h(x)
f  x  = ∣ x ∣ ,
g  x  = ∣ x ∣  1,
h  x = − ∣ x ∣
x − Achse


f  x  = ∣ x ∣
h  x  = − ∣ x ∣

OA
f  x  = ∣ x ∣
7


g  x  = ∣ x ∣  1
Precalculus
Root functions
y
g(x)
h(x)
h(x)
f(x)
x
Fig. 7: Root functions f (x), g(x) and h(x)
f  x = ∣ x ∣ ,
g  x  = ∣ x ∣  2 ,
D  f  x = D g  x  = ℝ ,
8
h  x = ∣ x ∣ − 2
D h  x = ( − ∞ , − 2 ] ∪ [ 2, ∞ )
Precalculus
Root functions: exercises 1,2
Exercise 1:
Describe how the graph of the function f (x)
is transformed to that of g(x)
a ) f x = x ,
gx = x  3  1
b ) f x = x ,
g  x  = 1.5  x  3
Exercise 2:
Plot the following functions
f  x  = 2 − x ,
9-A
g  x =  x  2  1
Precalculus
Root functions: solution 1a
y
g(x)
h(x)
f(x)
B
A
x
O
Fig. 8­1: Root functions f (x), g(x) and h(x)

OA
f x =
9-1
x



AB
hx =
x
3


g x =
x  3
1
Precalculus
Root functions: solution 1b
y
g(x)
h(x)
f(x)
A
x
O
Fig. 8­2: Root functions f (x), g(x) and h(x)

OA
f x =
9-2
x


× 1.5
hx =
x
3


g  x  = 1.5  x  3
Precalculus
Root functions: solution 2
g(x)
h(x)
f(x)
B
A
Fig. 9: Root functions f (x), g(x) and h(x)
f  x  = 2 − x
AB
hx =
9-3
x  2


g x = x  2  1
Precalculus