Inverse Mapping Worksheet Inverse Mapping
Absolute Value & Quadratic Analysis
NAME:
DATE:
1
Absolute Value Reflection
3 -2 -2 2 x y f(x) y = x
A) Graph the inverse relation on the coordinate plane.
Map key points using (x, y) → (y, x).
B) Intercept Analysis
Original f(x)
x-int:
y-int:
Inverse Relation
x-int:
y-int:
C) Is the inverse relation also a function? Justify your reasoning.
2
Quadratic Reflection
5 -4 2 4 x y g(x) y = x
A) Graph the inverse relation on the coordinate plane.
Determine the vertex and intercepts of the inverse.
B) Intercept Analysis
Original g(x)
x-int:
y-int:
Inverse Relation
x-int:
y-int:
C) Is the inverse relation also a function? Justify your reasoning.
Inverse Mapping Answer Key Answer Key
Inverse Mapping & Intercept Analysis
TEACHER RESOURCE
1
Absolute Value Mapping
f⁻¹(x)
B) Intercepts
Original \(f(x)\):
x-intercept: None
y-intercept: (0, 3)
Inverse Relation:
x-intercept: (3, 0)
y-intercept: None
C) Justification
NO. The inverse is not a function. When we reflect the V-shape across \(y=x\), it opens to the right. This means for most x-values (like \(x=3\)), there are two corresponding y-values. It fails the Vertical Line Test.
2
Quadratic Reflection
g⁻¹(x)
B) Intercepts
Original \(g(x)\):
x-intercept: None
y-intercept: (0, 5)
Inverse Relation:
x-intercept: (5, 0)
y-intercept: None
C) Justification
NO. The inverse fails the Vertical Line Test. Since the original quadratic function is not one-to-one (it fails the Horizontal Line Test), its reflection across \(y=x\) will have two outputs for a single input (e.g., \(x=5\) gives \(y=0\) and \(y=4\)).
Inverse Mapping Answer Key Answer Key
Inverse Mapping & Analysis
TEACHER KEY
1
Absolute Value Mapping
f-inv(x)
B) Intercepts
Original f(x):
x-int: None | y-int: (0, 3)
Inverse:
x-int: (3, 0) | y-int: None
C) Justification
No. The inverse fails the Vertical Line Test . For a single input (like x = 4), there are two different outputs. This occurs because the original function is not one-to-one.
2
Quadratic Reflection
g-inv(x)
B) Intercepts
Original g(x):
x-int: None | y-int: (0, 5)
Inverse:
x-int: (5, 0) | y-int: None
C) Justification
No. The inverse fails the Vertical Line Test . Parabolic curves opening horizontally have two y-values for each x-value in their domain (except the vertex), meaning they are not functions.
Inverse Mapping Answer Key Answer Key
Inverse Mapping Solutions
TEACHER USE
1
Absolute Value Mapping
f-inv(x)
B) Intercepts
Original f(x):
x-int: None | y-int: (0, 3)
Inverse:
x-int: (3, 0) | y-int: None
C) Justification
No. The inverse is not a function because it fails the Vertical Line Test . A single x-value (e.g., x=4) corresponds to two y-values (y=1 and y=-5). This is because the original function fails the Horizontal Line Test.
2
Quadratic Reflection
g-inv(x)
B) Intercepts
Original g(x):
x-int: None | y-int: (0, 5)
Inverse:
x-int: (5, 0) | y-int: None
C) Justification
No. The inverse is not a function. Since the original quadratic function is not one-to-one (fails HLT), its reflection across y=x is a horizontal parabola that fails the Vertical Line Test .
Inverse Mapping Worksheet Inverse Mapping
Absolute Value & Quadratic Relations
NAME:
DATE:
1
Absolute Value Mapping
0 2 4 -2 -4 2 4 -2 -4 x y y = x f(x)
A) Graph the inverse relation on the grid.
Reflection strategy: swap the coordinates (x, y) → (y, x).
B) Intercept Analysis
Original f(x)
x-int:
y-int:
Inverse Relation
x-int:
y-int:
C) Is the inverse relation also a function? Justify your reasoning.
2
Quadratic Mapping
0 2 4 -2 2 4 -2 x y g(x)
A) Graph the inverse relation on the grid.
Identify key points (vertex, intercepts) of g(x) first.
B) Intercept Analysis
Original g(x)
x-int:
y-int:
Inverse Relation
x-int:
y-int:
C) Is the inverse relation also a function? Justify your reasoning.
