One-to-One Function - Examples & Theorems

Question 1: True Statements about Functions (from file image_c693e9.png)

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

• If any horizontal line intersects the graph of a function $f$ in at most one point, then $f$ is one-to-one
• If any horizontal line intersects the graph of a function $f$ in at least one point, then $f$ is one-to-one.
• If $f$ is a decreasing function, then $f$ is not one-to-one.
• If $f$ is an increasing function, then $f$ is one-to-one.

Detailed Solution:

• First statement: This is the precise definition of the Horizontal Line Test. TRUE.
• Second statement: This is FALSE. This statement is not a valid test. For example, a horizontal line intersects the parabola $f(x)=x^2$ (which is not one-to-one) at two points.
• Third statement: This is FALSE. A function that is always decreasing will never have the same y-value twice, so it will always pass the Horizontal Line Test and therefore is one-to-one.
• Fourth statement: This is TRUE. A function that is always increasing will never have the same y-value twice. It always passes the Horizontal Line Test and is therefore one-to-one.

Final Answer: The true statements are:

• If any horizontal line intersects the graph of a function $f$ in at most one point, then $f$ is one-to-one
• If $f$ is an increasing function, then $f$ is one-to-one.

Question 2: Identifying One-to-One Functions (from file image_c693e9.png)

The Question: If $f: \mathbb{R} \to \mathbb{R}$, then which of the following functions is(are) one-to-one?

• $f(x) = |x+1| - 20$
• $f(x) = x^2 + 4x$
• $f(x) = x^3 + 15$
• $f(x) = x^3 - 5x^2 + 2x + 8$

Detailed Solution:

We can test each function by considering its graph and the Horizontal Line Test.

• $f(x) = |x+1| - 20$: This is an absolute value function, which has a “V” shape. A V-shaped graph will always fail the Horizontal Line Test (a horizontal line can cut through both arms of the V). This is not one-to-one.
• $f(x) = x^2 + 4x$: This is a quadratic function. Its graph is a parabola, which is a “U” shape. A U-shaped graph will always fail the Horizontal Line Test. This is not one-to-one.
• $f(x) = x^3 + 15$: The graph of $y=x^3$ is a curve that is always increasing. Adding 15 simply shifts the entire graph up by 15 units. It remains an always-increasing function. Therefore, it will pass the Horizontal Line Test. This function is one-to-one.
• $f(x) = x^3 - 5x^2 + 2x + 8$: This is a cubic function. Unlike $x^3$, this one has multiple terms that can create “hills and valleys” (local maximums and minimums). A cubic function with turning points will fail the Horizontal Line Test. We can check for turning points by seeing if its derivative, $f’(x) = 3x^2 - 10x + 2$, can equal zero for two different x-values. The discriminant of this derivative is $(-10)^2 - 4(3)(2) = 100 - 24 = 76 > 0$, which means there are two turning points. Therefore, this function is not one-to-one.

Final Answer: The only one-to-one function listed is $f(x) = x^3 + 15$.

Question 3: Identifying Decreasing Functions (from files image_c693e9.png, image_c69371.png)

The Question: Which of the following functions is(are) decreasing?

Detailed Solution:

A function is decreasing if its y-value always gets smaller as the x-value gets larger (the graph always goes downhill from left to right).

• $f(x) = 4x^2 + 4x + 1$: This is a parabola that opens upwards. It decreases to the left of its vertex but increases to the right. It is not a decreasing function over its entire domain.
• $f(x) = 7 - 3x$: This is a straight line with a slope of -3. A line with a negative slope is always going downhill. This is a decreasing function.
• $f(x) = -2x^3$: The basic function $y=x^3$ is always increasing. The negative sign in front, $-2$, reflects the graph across the x-axis, turning it into a function that is always going downhill. This is a decreasing function.
• $f(x) = \frac{1}{x}$: This function’s graph has two separate branches. While each branch is decreasing on its own interval, the function as a whole is not. For example, let $x_1 = -1$ and $x_2 = 1$. We have $x_1 < x_2$, but $f(x_1) = -1$ and $f(x_2) = 1$, so $f(x_1) < f(x_2)$. This violates the definition of a decreasing function. It is not a decreasing function over its entire domain.

Final Answer: The decreasing functions are:

• $f(x) = 7 - 3x$
• $f(x) = -2x^3$

Question 4: Identifying Incorrect Statements (from file image_c69371.png)

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

Detailed Solution:

We are looking for the statements that are false.

• “The function $f(x) = \frac{4x+3}{3x-5}$ for all $x \in \mathbb{R} \setminus {\frac{5}{3}}$ is one-to-one.”

  • This is a rational function whose graph is a hyperbola. Hyperbolas pass the Horizontal Line Test over their domain. This statement is TRUE.

• “The function $f(x) = mx+c$, for all $x \in \mathbb{R}$ (where $m \in \mathbb{R} \setminus {0}$ and $c \in \mathbb{R}$) is one-to-one.”

  • Since $m \neq 0$, this is a non-horizontal straight line. Any slanted line passes the Horizontal Line Test. This statement is TRUE.

• “A quadratic function having two distinct roots is always one-to-one.”

  • A quadratic function’s graph is a parabola. If it has two distinct roots, it crosses the x-axis twice. A horizontal line drawn between the vertex and the x-axis will intersect the parabola twice. Therefore, it always fails the Horizontal Line Test. This statement is FALSE.

• “A quadratic function having equal roots can be one-to-one.”

  • “Equal roots” means the parabola’s vertex is on the x-axis. It is still a U-shaped parabola. Any horizontal line drawn above (or below) the vertex will still intersect it twice. It always fails the Horizontal Line Test. This statement is FALSE.

Final Answer: The incorrect statements are:

• A quadratic function having two distinct roots is always one-to-one.
• A quadratic function having equal roots can be one-to-one.