SME
Grade 9 Math · Problem type
Domain and range
Use this for “f(x) = −2x + 5 for −1 ≤ x ≤ 4. What is its range?”
Chart
Domain: [−1, 4]
This is an example. Type your own values to start a new problem.
f(x)
=
m
x
+
b
for
x₁
≤
x
≤
x₂
range:
yₘᵢₙ
≤
f(x)
≤
yₘₐₓ
y₁
Output at x₁, f(x₁)
worked out (not typed)
y₂
Output at x₂, f(x₂)
worked out (not typed)
Formulas
y₁
=
m
×
x₁
+
b
7 = −2 × (−1) + 5
y₂
=
m
×
x₂
+
b
−3 = −2 × 4 + 5
yₘᵢₙ
=
the
smaller
of
y₁
and
y₂
−3 = the smaller of 7 and −3
yₘₐₓ
=
the
larger
of
y₁
and
y₂
7 = the larger of 7 and −3
Assumptions
•
The domain is the inputs allowed; the range is the outputs they give.
•
A line is highest and lowest at the ends of its domain.
•
Write each as an inequality, least first: −3 ≤ y ≤ 7.
Step-by-step
WE KNOW
m = −2, b = 5, x₁ = −1, x₂ = 4
FIND
output at x₁, f(x₁) (y₁), output at x₂, f(x₂) (y₂), least output (yₘᵢₙ), greatest output (yₘₐₓ)
Step 1 · Find output at x₁, f(x₁): y₁
Use y₁ = m × x₁ + b
The output at the left end of the domain.
y₁
=
−2
×
(−1)
+
5
y₁
=
2
+
5
y₁
=
7
Step 2 · Find output at x₂, f(x₂): y₂
Use y₂ = m × x₂ + b
The output at the right end.
y₂
=
−2
×
4
+
5
y₂
=
−8
+
5
y₂
=
−3
Step 3 · Find least output (yₘᵢₙ)
Use yₘᵢₙ = the smaller of y₁ and y₂
A line only rises or only falls, so the least output is at one end.
yₘᵢₙ = the smaller of 7 and −3
yₘᵢₙ = −3
Step 4 · Find greatest output (yₘₐₓ)
Use yₘₐₓ = the larger of y₁ and y₂
And the greatest output at the other end.
yₘₐₓ = the larger of 7 and −3
yₘₐₓ
=
7
→
range:
−3
≤
f
(x)
≤
7
Check
Put the numbers back in. Both sides should match.
7 = −2 × (−1) + 5 ✓
−3 = −2 × 4 + 5 ✓
−3 = the smaller of 7 and −3 ✓
7 = the larger of 7 and −3 ✓
Topic check
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