One-to-One Function | Definition & Tests

Question 1: The Purpose of the Line Tests (from file image_d08bd4.png)

The Question: Which of the following statements is(are) TRUE?

• The Vertical line test is used to find whether the given function is one-to-one or not.
• The Horizontal line test is used to find whether the given function is one-to-one or not.
• If for one value of $x$ in domain gives more than one $f(x)$, then $f$ is one-to-one function.
• If for more than one value of $x$ in domain gives one $f(x)$, then $f$ is one-to-one function.

Detailed Solution:

Let’s evaluate each statement based on our core concepts.

• First statement: This is FALSE. The Vertical Line Test determines if a relation is a function in the first place, not if it’s one-to-one.
• Second statement: This is TRUE. This is the exact purpose of the Horizontal Line Test.
• Third statement: This is FALSE. If one value of $x$ gives more than one value of $f(x)$, the relation is not a function at all.
• Fourth statement: This is FALSE. If more than one value of $x$ gives the same $f(x)$ (e.g., $f(2)=4$ and $f(-2)=4$), this is the definition of a function that is not one-to-one.

Final Answer: The only true statement is “The Horizontal line test is used to find whether the given function is one-to-one or not.”

Question 2: Failing the Horizontal Line Test (from file image_d088d1.png)

The Question: Which of the following graph fails the Horizontal line test?

Core Concept: A graph fails the Horizontal Line Test if you can draw at least one horizontal line that crosses the graph in more than one place.

Detailed Solution:

• Graph 1 (Cubic-like function): A horizontal line can be drawn that intersects the graph in three places. It fails the test.
• Graph 2 (Linear function): Any horizontal line will cross this graph exactly once. It passes the test.
• Graph 3 (Square root-like function): Any horizontal line will cross this graph at most once. It passes the test.
• Graph 4 (Parabola): A horizontal line drawn above the vertex will intersect the graph in two places. It fails the test.

Final Answer: The first graph and the fourth graph fail the Horizontal Line Test.

Question 3: True Statements about Functions (from file image_d08857.png)

The Question: Which of the following statements is(are) TRUE?

• One-to-one functions never fail the Horizontal line test.
• One-to-one functions may sometime fail the Horizontal line test.
• No function should fail the Horizontal line test
• There are some functions that fail the Vertical line test.

Detailed Solution:

• First statement: This is TRUE. Passing the Horizontal Line Test is the graphical definition of a one-to-one function.
• Second statement: This is FALSE. If a function fails the HLT, it is by definition not one-to-one.
• Third statement: This is FALSE. Many valid functions are not one-to-one (e.g., $y=x^2$) and therefore fail the HLT.
• Fourth statement: This is FALSE. This is a trick statement. If a graph fails the Vertical Line Test, it is not a function by definition. Therefore, a “function” cannot fail the Vertical Line Test.

Final Answer: The only true statement is “One-to-one functions never fail the Horizontal line test.”

Question 4: False Statements about One-to-One Functions (from file image_d08857.png)

The Question: Which of the following statements is(are) FALSE?

Core Concept: The question asks for the false statement. We must check the algebraic definition of a one-to-one function and its link to being reversible (invertible).

Detailed Solution:

• “A function $f: X \to Y$ is called one-to-one if, if $f(x_1) = f(x_2) \in Y$, then $x_1 = x_2$.”
  • This is the formal definition of a one-to-one function. It means that if the outputs are the same, the inputs must have been the same. This statement is TRUE.
• “One-to-one functions are not always reversible on their range.”
  • This is FALSE. A function is reversible (meaning its inverse is also a function) if and only if it is one-to-one. This is a fundamental property.
• “If a function fails the Horizontal line test, then it is not reversible.”
  • If a function fails the HLT, it is not one-to-one. If it’s not one-to-one, it is not reversible. This statement is TRUE.
• “A function $f: X \to Y$ is called one-to-one if, if $f(x_1) \neq f(x_2) \in Y$, then $x_1 \neq x_2$.”
  • This is the “contrapositive” of the formal definition. It means that if the inputs are different, the outputs must be different. This is an equally valid definition. This statement is TRUE.

Final Answer: The false statement is “One-to-one functions are not always reversible on their range.”

Question 5: Identifying a Non-Function (from file image_d08857.png)

The Question: Suppose $f:[0, \infty) \to \mathbb{R}$. Which of the following is not a function?

Core Concept: A relation is not a function if a single input value $x$ can produce more than one output value $y$.

Detailed Solution:

Let’s examine each option.

• $f(x) = 4x+3$: This is a linear equation. For any input $x$, there is only one possible output $y$. This is a function.
• $f(x) = \pm\sqrt{x} + 1$: The ± symbol is the key. It means “plus or minus”. Let’s test an input, for example, $x=4$.
  • One output is $y = +\sqrt{4} + 1 = 2 + 1 = 3$.
  • Another output is $y = -\sqrt{4} + 1 = -2 + 1 = -1$.
  • Since the single input $x=4$ produces two different outputs (3 and -1), this is not a function.
• $f(x) = x^2 + 4x + 30$: This is a quadratic equation. For any input $x$, there is only one possible output $y$. This is a function.
• $f(x) = 100$: This is a constant function. For any input $x$, the output is always 100. This is a function.

Final Answer: The relation that is not a function is $f(x) = \pm\sqrt{x} + 1$.