Question

Please help with this

Answer 1

Answer:

  (a)  Horizontal line test

Step-by-step explanation:

A relation that is a function already passes the vertical line test. If it has an inverse function, that function must also pass the vertical line test.

Inverse relation

The inverse relation will be the same graph reflected across the line y=x. That is, points that are horizontally aligned on the relation graph become vertically aligned on the inverse relation graph.

Inverse function

If the inverse relation is to be a function, then there must be no points on its graph that are vertically aligned. This requirement translates to the requirement that there be no horizontally-aligned points on the graph of the original relation. The "horizontal line test" checks for horizontally aligned points. If there are none, the test is said to pass.

A function will have an inverse function if it passes the Horizontal Line Test.

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Additional comment

The attached graph shows a function (red curve) that does not pass the horizontal line test. Its inverse relation is the green curve, which you can see is not a function. It does not pass the vertical line test.

The line y=x is shown (dashed orange) so you can see the symmetry of the function and its inverse.