In a function f(x), if f(a) = f(b) for a ≠ b, what can be inferred about the fun

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Question: In a function f(x), if f(a) = f(b) for a ≠ b, what can be inferred about the function?

Options:

Correct Answer: The function is constant.

Solution:

If f(a) = f(b) for a ≠ b, it indicates that the function is not one-to-one, which means it does not pass the horizontal line test.

In a function f(x), if f(a) = f(b) for a ≠ b, what can be inferred about the function?

In a function f(x), if f(a) = f(b) for a ≠ b, what can be inferred about the function?

Step 1: Understand what f(a) = f(b) means. It means that when you plug in 'a' into the function f, you get the same result as when you plug in 'b'.
Step 2: Recognize that 'a' and 'b' are different values (a ≠ b). This means that two different inputs give the same output.
Step 3: Know that a function is called 'one-to-one' if every input gives a unique output. Since f(a) = f(b) for different inputs, this function is not one-to-one.
Step 4: Learn about the horizontal line test. If you can draw a horizontal line that intersects the graph of the function at more than one point, the function is not one-to-one.
Step 5: Conclude that since f(a) = f(b) for a ≠ b, the function does not pass the horizontal line test and is therefore not one-to-one.

Injective Function – A function is injective (one-to-one) if different inputs produce different outputs.
Horizontal Line Test – A graphical method to determine if a function is one-to-one; if any horizontal line intersects the graph more than once, the function is not one-to-one.