Understanding Functions in Mathematics

A mathematical function is a special type of relation where each input value (from the domain) corresponds to exactly one output value (in the range). In simpler terms, for any given 'x' value, there should be only one 'y' value. We can visually test if an equation represents a function using the vertical line test. If any vertical line intersects the graph of the equation at more than one point, then the equation does not represent a function.

Equation Analysis for Function Representation

Let's analyze each given option to determine which one represents a function.

• Equation: $\lvert x \rvert + \lvert y \rvert = 2$

  To check if this is a function, let's consider an example:

  • If we set $x = 0$, the equation becomes $\lvert 0 \rvert + \lvert y \rvert = 2$, which simplifies to $\lvert y \rvert = 2$.
  • This means $y = 2$ or $y = -2$.

  Since one input value ($x=0$) results in two distinct output values ($y=2$ and $y=-2$), this equation does not represent a function. Graphically, this equation forms a square rotated by 45 degrees, and a vertical line can intersect it at two points.

• Equation: $\lvert x + y \rvert = 2$

  This absolute value equation can be split into two separate linear equations:

  • $x + y = 2 \implies y = -x + 2$
  • OR
  • $x + y = -2 \implies y = -x - 2$

  Let's pick an input value, for example, $x = 0$:

  • From $y = -x + 2$, if $x = 0$, then $y = -0 + 2 = 2$.
  • From $y = -x - 2$, if $x = 0$, then $y = -0 - 2 = -2$.

  Here again, one input value ($x=0$) corresponds to two different output values ($y=2$ and $y=-2$). Therefore, this equation does not represent a function. Graphically, it represents two parallel lines, failing the vertical line test.

• Equation: $\lvert y \rvert = x^2 + \sin x$

  Let's examine this equation. For any value of $x$ where $x^2 + \sin x$ is a positive number, there will be two possible values for $y$ due to the absolute value around $y$.

  • For example, let's choose $x = 1$. The right side of the equation becomes $1^2 + \sin(1) = 1 + \sin(1)$.
  • Since $\sin(1)$ is approximately $0.841$, $1 + \sin(1)$ is approximately $1.841$, which is a positive number.
  • So, $\lvert y \rvert = 1 + \sin(1)$, which implies $y = +(1 + \sin(1))$ or $y = -(1 + \sin(1))$.

  As one input value ($x=1$) leads to two distinct output values, this equation does not represent a function.

• Equation: $\lvert x \rvert^2 - x$

  Let's consider this expression as $y = \lvert x \rvert^2 - x$.

  • We know that for any real number $x$, the square of its absolute value, $\lvert x \rvert^2$, is always equal to $x^2$. This is because squaring a number always results in a non-negative value, whether the original number was positive or negative, just like the absolute value makes it positive before squaring.
  • So, the equation can be rewritten as $y = x^2 - x$.
  • This is a standard quadratic equation, which represents a parabola opening upwards. For every single real value of $x$ that you input into this equation, there will be exactly one unique real value of $y$ as an output.

  Therefore, this equation successfully passes the vertical line test and represents a function.

Conclusion

Based on our analysis, only the equation derived from the expression $\lvert x \rvert^2 - x$ satisfies the definition of a function, because for every input $x$, there is exactly one output $y$.