A2 Lesson 20: Explore: The Horizontal Line Test Solution

The Horizontal Line Test Solution

The horizontal line test (HLT) is a visual representation that determines if the inverse of a graph on the coordinate plane is a function by running a horizontal line ↔ across the graph. 
If the horizontal line touches more than one point at a time, the inverse of the graph is not a function. 
The HLT also determines if a function is one-to-one . 
A one-to-one function has one output for each input. (In other words, the domain AND range values are both unique for a one-to-one function.)

Note

A graph can still be made on a coordinate plane even when a function does not exist.

Example 4

For the given graph:

- Name the domain and range for the graph and its inverse.
- Explain whether or not the graph represents a function. 
- If the graph is a function, determine if it is one-to-one.
- If the graph is a function, determine if the inverse is also a function.

Given
Domain: $\{x | x \in R\}$
Range: $\{y | y \in R, y \geq - 2\}$  Inverse
Domain: $\{x | x \in R, x \geq - 2\}$
Range: $\{y | y \in R\}$  The parabola is a function because it passes the VLT. However, it is not one-to-one  because it fails the HLT. This also means that the inverse is not a function.

You can plot the inverse on the coordinate plane to help see if the inverse is a function using the VLT and determine the domain and range for the inverse if needed.

For MM:

Example 5

For the given graph:

Name the domain and range for the graph and its inverse.

Explain whether or not the graph represents a function.

If the graph is a function, determine if it is one-to-one.

If the graph is a function, determine if the inverse is also a function.

Given
Domain: “`MathML
<math style=”font-family:Times New Roman;font-size:18px;” xmlns=”http://www.w3.org/1998/Math/MathML”><mstyle mathsize=”18px”><mfenced mathcolor=”#C15300″ open=”{” close=”}”><mrow><mi>x</mi><mo>|</mo><mi>x</mi><mo>∈</mo><mi>R</mi></mrow></mfenced></mstyle></math>
“`
Range: “`MathML
<math style=”font-family:Times New Roman;font-size:18px;” xmlns=”http://www.w3.org/1998/Math/MathML”><mstyle mathsize=”18px”><mfenced mathcolor=”#C15300″ open=”{” close=”}”><mrow><mi>y</mi><mo>|</mo><mi>y</mi><mo>∈</mo><mi>R</mi></mrow></mfenced></mstyle></math>
“`

Inverse
Domain: “`MathML
<math style=”font-family:Times New Roman;font-size:18px;” xmlns=”http://www.w3.org/1998/Math/MathML”><mstyle mathsize=”18px”><mfenced mathcolor=”#C15300″ open=”{” close=”}”><mrow><mi>x</mi><mo>|</mo><mi>x</mi><mo>∈</mo><mi>R</mi></mrow></mfenced></mstyle></math>
“`
Range: “`MathML
<math style=”font-family:Times New Roman;font-size:18px;” xmlns=”http://www.w3.org/1998/Math/MathML”><mstyle mathsize=”18px”><mfenced mathcolor=”#C15300″ open=”{” close=”}”><mrow><mi>y</mi><mo>|</mo><mi>y</mi><mo>∈</mo><mi>R</mi></mrow></mfenced></mstyle></math>
“`

The (cubic) graph is a function because it passes the VLT. The graph also passes the HLT. This means that the given function is one-to-one, and its inverse is also a function.

Naming the graph is good practice to retain recognition of equations graphically.

For MM

Example 6

For the given graph:

Name the domain and range for the graph and its inverse.

Explain whether or not the graph represents a function.

If the graph is a function, determine if it is one-to-one.

If the graph is a function, determine if the inverse is also a function.

Range: “`MathML
<math style=”font-family:Times New Roman;font-size:18px;” xmlns=”http://www.w3.org/1998/Math/MathML”><mstyle mathsize=”18px”><mfenced mathcolor=”#C15300″ open=”{” close=”}”><mrow><mi>y</mi><mo>|</mo><mi>y</mi><mo>∈</mo><mi>R</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>≥</mo><mn>1</mn></mrow></mfenced></mstyle></math>
“`

Inverse
Domain: “`MathML
<math style=”font-family:Times New Roman;font-size:18px;” xmlns=”http://www.w3.org/1998/Math/MathML”><mstyle mathsize=”18px”><mfenced mathcolor=”#C15300″ open=”{” close=”}”><mrow><mi>x</mi><mo>|</mo><mi>x</mi><mo>∈</mo><mi>R</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mn>1</mn></mrow></mfenced></mstyle></math>
“`
Range: “`MathML
<math style=”font-family:Times New Roman;font-size:18px;” xmlns=”http://www.w3.org/1998/Math/MathML”><mstyle mathsize=”18px”><mfenced mathcolor=”#C15300″ open=”{” close=”}”><mrow><mi>y</mi><mo>|</mo><mi>y</mi><mo>∈</mo><mi>R</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>≥</mo><mo>–</mo><mn>4</mn></mrow></mfenced></mstyle></math>
“`

The (square root) graph is a function because it passes the VLT. The graph also passes the HLT. This means that the given function is one-to-one, and its inverse is also a function.

For MM