MAT 500 Test 3 Review

Vocabulary practice flashcards covering piecewise functions, polynomial inequalities, polynomial characteristics (degree and intercepts), and rational function properties (intercepts and domain) from the MAT 500 Test 3 review sheet.

Piecewise Function $f(x)$ (Problem 1a)

A piecewise-defined function given by $f(x) = -2x - 5$ for $x \le -4$, $f(x) = x - 7$ for $-4 < x \le 2$, and $f(x) = 3x - 4$ for $x > 2$.

Piecewise Function $f(x)$ (Problem 1b)

A piecewise-defined function given by $f(x) = 2x + 13$ for $x < -3$, and $f(x) = 0.5x - 4$ for $x \ge -3$.

Piecewise Function $f(x)$ (Problem 1c)

A piecewise-defined function given by $f(x) = -0.5x + 2$ for $x < 2$, and $f(x) = x - 3$ for $x \ge 2$.

Solution to $(x + 3)(x - 5)(x - 11) < 0$

$x < -3$ or $5 < x < 11$

Solution to $(x - 2)(x + 5)^2(x - 7) > 0$

$x < -5$ or $-5 < x < 2$ or $x > 7$

Solution to $-2x^2 - 6x + 20 \le 0$

$x \le -5$ or $x \ge 2$

Solution to $x^3 - 7x^2 - 18x < 0$

$x < -2$ or $0 < x < 9$

Solution to $2x^2 + 17x - 20 \le x^2 + 2x - 74$

$$-9 \le x \le -6$$

Polynomial Properties of $f(x) = x^2 + 5x - 24$

Degree: $2$; y-intercept: $(0, -24)$; x-intercepts: $(-8, 0)$ and $(3, 0)$.

Polynomial Properties of $f(x) = 2(x - 1)(x + 5)(x^2 - 8x + 40)$

Degree: $4$; y-intercept: $(0, -400)$; x-intercepts: $(1, 0)$ and $(-5, 0)$.

Polynomial Properties of $f(x) = -4(x + 2)(x - 4)(x + 5)(x + 8)^2$

Degree: $5$; y-intercept: $(0, 10240)$; x-intercepts: $(-2, 0)$, $(4, 0)$, $(-5, 0)$, and $(-8, 0)$ (with multiplicity $2$).

Polynomial Properties of $f(x) = -2x(x - 3)(x + 1)$

Degree: $3$; y-intercept: $(0, 0)$; x-intercepts: $(0, 0)$, $(3, 0)$, and $(-1, 0)$.

Polynomial Properties of $f(x) = x^4 + 2x^3 - 3x^2$

Degree: $4$; y-intercept: $(0, 0)$; x-intercepts: $(0, 0)$ (with multiplicity $2$), $(-3, 0)$, and $(1, 0)$.

Polynomial Properties of $f(x) = 5(x - 5)(x + 1)^2(x + 3)$

Degree: $4$; y-intercept: $(0, -75)$; x-intercepts: $(5, 0)$, $(-3, 0)$, and $(-1, 0)$ (with multiplicity $2$).

Rational Function Properties of $f(x) = \frac{2(x - 1)(x + 5)}{x + 3}$

y-intercept: $(0, -\frac{10}{3})$; x-intercepts: $(1, 0)$ and $(-5, 0)$; Domain: $x \neq -3$.

Rational Function Properties of $f(x) = \frac{(x - 4)(x + 7)}{(x + 3)(x - 5)(x + 2)}$

y-intercept: $(0, \frac{14}{15})$; x-intercepts: $(4, 0)$ and $(-7, 0)$; Domain: $x \neq -3, 5, -2$.

Rational Function Properties of $f(x) = \frac{4(x - 5)(x + 2)}{(x + 3)(x - 3)}$

y-intercept: $(0, \frac{40}{9})$; x-intercepts: $(5, 0)$ and $(-2, 0)$; Domain: $x \neq 3, -3$.

Rational Function Properties of $f(x) = \frac{(x - 6)(x + 2)}{(x + 2)(x + 3)}$

y-intercept: $(0, -2)$; x-intercept: $(6, 0)$; Domain: $x \neq -3, -2$.

Rational Function Properties of $f(x) = \frac{3(x - 3)(x + 5)}{(x - 3)(x + 2)}$

y-intercept: $(0, \frac{15}{2})$; x-intercept: $(-5, 0)$; Domain: $x \neq 3, -2$.

Rational Function Properties of $f(x) = \frac{(x + 7)(x - 9)(x + 2)}{x - 7}$

y-intercept: $(0, 18)$; x-intercepts: $(-7, 0)$, $(9, 0)$, and $(-2, 0)$; Domain: $x \neq 7$.