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SAT II Math II : Solving Functions

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Example Question #1 : Solving Piecewise And Recusive Functions

Which of the following would be a valid alternative definition for the provided function?

$f(x) = \left | \frac{3}{4} x - \frac{7}{3} \right |$

Possible Answers:

$f(x) = \left\{\begin{matrix} - \frac{3}{4} x + \frac{7}{3} & \textrm{for }x < \frac{28}{9}\\ \\ \frac{3}{4} x - \frac{7}{3}& \textrm{for }x \ge \frac{28}{9} \end{matrix}\right.$

None of these

$f(x) = \left\{\begin{matrix} - \frac{3}{4} x + \frac{7}{3} & \textrm{for }x < -\frac{28}{9}\\ \\ \frac{3}{4} x - \frac{7}{3}& \textrm{for }x \ge - \frac{28}{9} \end{matrix}\right.$

$f(x) = \left\{\begin{matrix} \frac{3}{4} x - \frac{7}{3} & \textrm{for }x < \frac{28}{9}\\ \\ - \frac{3}{4} x + \frac{7}{3} & \textrm{for }x \ge \frac{28}{9} \end{matrix}\right.$

$f(x) = \left\{\begin{matrix} \frac{3}{4} x - \frac{7}{3} & \textrm{for }x < - \frac{28}{9}\\ \\ - \frac{3}{4} x + \frac{7}{3} & \textrm{for }x \ge - \frac{28}{9} \end{matrix}\right.$

Correct answer:

$f(x) = \left\{\begin{matrix} - \frac{3}{4} x + \frac{7}{3} & \textrm{for }x < \frac{28}{9}\\ \\ \frac{3}{4} x - \frac{7}{3}& \textrm{for }x \ge \frac{28}{9} \end{matrix}\right.$

Explanation:

The absolute value of an expression | a | is defined as follows:

| a | = a for $a \ge 0$

| a | = -a for a < 0

Therefore,

$f(x) = \left | \frac{3}{4} x - \frac{7}{3} \right | = \frac{3}{4} x - \frac{7}{3}$

if and only if

$\frac{3}{4} x - \frac{7}{3} \ge 0$ .

Solving this condition for :

$\frac{3}{4} x - \frac{7}{3} + \frac{7}{3} \ge 0 + \frac{7}{3}$

$\frac{3}{4} x \ge \frac{7}{3}$

$\frac{4}{3} \cdot \frac{3}{4} x \ge \frac{4}{3} \cdot \frac{7}{3}$

$x \ge \frac{28}{9}$

Therefore, $f(x) = \frac{3}{4} x - \frac{7}{3}$ for $x \ge \frac{28}{9}$ .

Similarly,

$f(x) = - \left (\frac{3}{4} x - \frac{7}{3} \right ) = - \frac{3}{4} x + \frac{7}{3}$ for $x < \frac{28}{9}$ .

The correct response is therefore

$f(x) = \left\{\begin{matrix} - \frac{3}{4} x + \frac{7}{3} & \textrm{for }x < \frac{28}{9}\\ \\ \frac{3}{4} x - \frac{7}{3}& \textrm{for }x \ge \frac{28}{9} \end{matrix}\right.$

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SAT II Math II : Solving Functions

Study concepts, example questions & explanations for SAT II Math II

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Example Question #1 : Solving Piecewise And Recusive Functions

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