Pre-Calculus - Determine if a Relation is a Function

Is the following relation of ordered pairs a function?

Yes

Cannot be determined

Explanation

A set of ordered pairs is a function if it passes the vertical line test.

Because there are no more than one corresponding value for any given value, the relation of ordered pairs IS a function.

Suppose we have the relation $\small (a,b)$ on the set of real numbers $\mathbb{R}$ whenever $\small a<b$ . Which of the following is true.

The relation is not a function because $\small (2,3)$ and $\small (2,4)$ both hold.

The relation is not a function because $\small (0,1)$ holds but $\small (1,0)$ does not.

The relation is a function because $\small (2,4)$ holds and $\small (3,4)$ also holds.

The relation is a function because for every $\small a$ , there is only one $\small b$ such that $\small (a,b)$ holds.

The relation is a function because every relation is a function, since that's how relations are defined.

Explanation

The relation is not a function because $\small (2,3)$ and $\small (2,4)$ hold. If it were a function, $\small (2,b)$ would hold only for one $\small b$ . But we know it holds for $\small b=3,4$ because $\small 2<3$ and $\small 2<4$ . Thus, the relation $\small (a,b)$ on the set of real numbers $\small \mathbb{R}$ is not a function.

Consider a family consisting of a two parents, Juan and Oksana, and their daughters Adriana and Laksmi. A relation $\small (a,b)$ is true whenever $\small b$ is the child of $\small a$ . Which of the following is not true?

Even if the two parents had only one daughter, the relation would not be a function.

If the two parents had only one daughter, the relation would be a function.

The relation is not a function because (Laksmi,Juan) and (Laksmi,Oksana) both hold.

(Laksmi,Adriana) does not hold because Adriana is not Laksmi's child.

(Adriana,Laksmi) does not hold because Laksmi is not Adriana's child and is a boy.

Explanation

The statement

"Even if the two parents had only one daughter, the relation would not be a function."

is not true because if they had only one daughter, say Adriana, then the only relations that would exist would be (Juan, Adriana) and (Oksana,Adriana), which defines a function.

What equation is perpendicular to $y = \frac{5}{2}x+12$ and passes throgh ?

$y = -\frac{2}{5}x + \frac{29}{5}$

$y = -\frac{4}{5}x + \frac{26}{5}$

$y = -\frac{2}{5}x + \frac{27}{5}$

$y = -\frac{5}{2}x + \frac{22}{5}$

$y = \frac{2}{5}x + \frac{21}{5}$

Explanation

First find the reciprocal of the slope of the given function.
$m_2 = -\frac{1}{m_1}$

$m_2 = -\frac{2}{5}$

The perpendicular function is:

$y = -\frac{2}{5}x+c$

Now we must find the constant, , by using the given point that the perpendicular crosses.

$5 = -\frac{2}{5}(2)+c$

solve for :

$c = \frac{29}{5}$

$y = -\frac{2}{5}x + \frac{29}{5}$

Which of the following relations is not a function?

$y = \sqrt{x}$

$y = x^{\frac{3}{2}}$

Explanation

The definition of a function requires that for each input (i.e. each value of ), there is only one output (i.e. one value of ). For , each value of corresponds to two values of (for example, when , both and are correct solutions). Therefore, this relation cannot be a function.

Which of the following expressions is not a function?

$y=\pm\sqrt{x^2-7x+5}$

Explanation

Recall that an expression is only a function if it passes the vertical line test. Test this by graphing each function and looking for one which fails the vertical line test. (The vertical line test consists of drawing a vertical line through the graph of an expression. If the vertical line crosses the graph of the expression more than once, the expression is not a function.)

Functions can only have one y value for every x value. The only choice that reflects this is:

$y=\pm\sqrt{x^2-7x+5}$