Let y = f(x) be a function with domain D = [−6, −2] and range R = [−10, −4].
Find the domain D and range R for each function. (Enter your answers using
interval notation
a) y = f(2x)
Solve for f(expression)
f(2x) = y
DOMAIN:
f(2x) = y
Make inequality of domain with expression in the middle,
and solve for x in the middle
-6 ≤ 2x ≤ -2
Divide all three sides by 2 to get x in the middle
-3 ≤ x ≤ -1
New domain = [-3, -1]
RANGE:
f(2x) = y
Make inequality of range with right side in the middle,
and solve for y in the middle
-10 ≤ y ≤ 4
y is already solved for in the middle, so
range = [-10, 4] (That is, the range is not changed.
----------------------------------------------------
b) y = f(x − 2) + 5
Solve for f(expression)
f(x - 2) = y - 5
DOMAIN:
f(x - 2) = y - 5
Make inequality of domain with expression in the middle,
and solve for x in the middle
-6 ≤ x - 2 ≤ -2
Add 2 to all three sides to get x in the middle
-4 ≤ x ≤ 0
New domain = [-4, 0]
RANGE:
f(x - 2) = y - 5
Make inequality of range with right side in the middle,
and solve for y in the middle
-10 ≤ y - 5 ≤ 4
Add 5 to all three sides to get y in the middle
-5 ≤ y ≤ 9
New range = [-5, 9]
----------------------------------------------------
c) y = f(x + 4) − 1
Solve for f(expression)
f(x + 4) = y + 1
DOMAIN:
f(x + 4) = y + 1
Make inequality of domain with expression in the middle,
and solve for x in the middle
-6 ≤ x + 4 ≤ -2
Add -4 to all three sides to get x in the middle
-10 ≤ x ≤ -6
New domain = [-10, -6]
RANGE:
f(x + 4) = y + 1
Make inequality of range with right side in the middle,
and solve for y in the middle
-10 ≤ y + 1 ≤ 4
Add -1 to all three sides to get y in the middle
-11 ≤ y ≤ 3
New range = [-11, 3]
----------------------------------------------------
d) y = f(−x)
Solve for f(expression)
f(-x) = y
DOMAIN:
f(-x) = y
Make inequality of domain with expression in the middle,
and solve for x in the middle
-6 ≤ -x ≤ -2
Multiply all three sides by -1 to get x in the middle,
remembering that multiplying through an equality changes
the direction of the inequality
6 ≥ x ≥ 2, which is the same as
2 ≤ x ≤ 6
New domain = [2, 6]
RANGE:
f(-x) = y
Make inequality of range with right side in the middle,
and solve for y in the middle
-10 ≤ y ≤ 4
y is already solved for in the middle, so
range = [-10, 4] (That is, the range is not changed.
----------------------------------------------------
e) y = −f(x)
Solve for f(expression)
f(x) = -y
DOMAIN:
f(x) = -y
Make inequality of domain with expression in the middle,
and solve for x in the middle
-6 ≤ x ≤ -2
x is already solved for in the middle, so
domain = [-6, -2] (That is, the domain is not changed.
RANGE:
f(x) = -y
Make inequality of range with right side in the middle,
and solve for y in the middle
-10 ≤ -y ≤ 4
Multiply all three sides by -1 to get y in the middle,
remembering that multiplying through an equality changes
the direction of the inequality
10 ≥ y ≥ -4, which is the same as
-4 ≤ y ≤ 10
New range = [-4, 10]
----------------------------------------------------
y = |f(x)|
This one is different because we cannot solve for f(expression)
But we still make inequality of domain with expression in the middle,
and solve for x in the middle
-6 ≤ x ≤ -2
x is already solved for in the middle, so
domain = [-6, -2] (That is, the domain is not changed.)
The range is different. Whatever input value(s) of x causes output
value of -10 for f(x), will cause output value of |-10| = 10; and
whatever input value(s) of x causes output value of -4 for f(x), will
cause output value of |-4| = 4. So,
new range = [4, 10]
Edwin
