명제와 조건

Following is a statement of a theorem which can be proven using calculus or precalculus mathematics. For this theorem, $a$, $b$, and $c$ are real numbers.

• Theorem If $f$ is a quadratic function of the form $f\left( x \right) = ax^2 + bx + c$ and $a \lt 0$, then the function $f$ has a maximum value when $x = \dfrac{{ - b}}{{2a}}$.

Using only this theorem, what can be concluded about the functions given by the following formulas?

$F\left( x \right) = - x^4 + x^3 + 9$

Following is a statement of a theorem which can be proven using the quadratic formula. For this theorem, $a$, $b$, and $c$ are real numbers.

• Theorem If $f$ is a quadratic function of the form $f\left( x \right) = ax^2 + bx + c$ and $ac \lt 0$, then the function $f$ has two $x$-intercepts.

Using only this theorem, what can be concluded about the functions given by the following formulas?

$g\left( x \right) = - 8x^2 + 5x - 2$

Following is a statement of a theorem which can be proven using the quadratic formula. For this theorem, $a$, $b$, and $c$ are real numbers.

• Theorem If $f$ is a quadratic function of the form $f\left( x \right) = ax^2 + bx + c$ and $ac \lt 0$, then the function $f$ has two $x$-intercepts.

Using only this theorem, what can be concluded about the functions given by the following formulas?

$h\left( x \right) = - \dfrac{1}{3}x^2 + 3x$

Following is a statement of a theorem which can be proven using the quadratic formula. For this theorem, $a$, $b$, and $c$ are real numbers.

• Theorem If $f$ is a quadratic function of the form $f\left( x \right) = ax^2 + bx + c$ and $ac \lt 0$, then the function $f$ has two $x$-intercepts.

Using only this theorem, what can be concluded about the functions given by the following formulas?

$k\left( x \right) = 8x^2 - 5x - 7$

Following is a statement of a theorem which can be proven using the quadratic formula. For this theorem, $a$, $b$, and $c$ are real numbers.

• Theorem If $f$ is a quadratic function of the form $f\left( x \right) = ax^2 + bx + c$ and $ac \lt 0$, then the function $f$ has two $x$-intercepts.

Using only this theorem, what can be concluded about the functions given by the following formulas?

$j\left( x \right) = - \dfrac{{71}}{{99}}x^2 + 210$

Following is a statement of a theorem which can be proven using the quadratic formula. For this theorem, $a$, $b$, and $c$ are real numbers.

• Theorem If $f$ is a quadratic function of the form $f\left( x \right) = ax^2 + bx + c$ and $ac \lt 0$, then the function $f$ has two $x$-intercepts.

Using only this theorem, what can be concluded about the functions given by the following formulas?

$f\left( x \right) = - 4x^2 - 3x + 7$

Following is a statement of a theorem which can be proven using the quadratic formula. For this theorem, $a$, $b$, and $c$ are real numbers.

• Theorem If $f$ is a quadratic function of the form $f\left( x \right) = ax^2 + bx + c$ and $ac \lt 0$, then the function $f$ has two $x$-intercepts.

Using only this theorem, what can be concluded about the functions given by the following formulas?

$F\left( x \right) = - x^4 + x^3 + 9$

Following is a statement of a theorem about certain cubic equations. For this theorem, $b$ represents a real number.

• Theorem A If $f$ is a cubic function of the form $f(x) = x^3 - x + b$ and $b > 1$, then the function $f$ has exactly one $x$-intercept.

Following is another theorem about $x$-intercepts of functions:

• Theorem B If $f$ and $g$ are functions with $g(x) = k \cdot f(x)$, where $k$ is a nonzero real number, then $f$ and $g$ have exactly the same $x$-intercepts.

Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas?

$f(x) = x^3 - x + 7$

Following is a statement of a theorem about certain cubic equations. For this theorem, $b$ represents a real number.

• Theorem A If $f$ is a cubic function of the form $f(x) = x^3 - x + b$ and $b > 1$, then the function $f$ has exactly one $x$-intercept.

Following is another theorem about $x$-intercepts of functions:

• Theorem B If $f$ and $g$ are functions with $g(x) = k \cdot f(x)$, where $k$ is a nonzero real number, then $f$ and $g$ have exactly the same $x$-intercepts.

Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas?

$g(x) = x^3 + x + 7$

Following is a statement of a theorem about certain cubic equations. For this theorem, $b$ represents a real number.

• Theorem A If $f$ is a cubic function of the form $f(x) = x^3 - x + b$ and $b > 1$, then the function $f$ has exactly one $x$-intercept.

Following is another theorem about $x$-intercepts of functions:

• Theorem B If $f$ and $g$ are functions with $g(x) = k \cdot f(x)$, where $k$ is a nonzero real number, then $f$ and $g$ have exactly the same $x$-intercepts.

Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas?

$h(x) = -x^3 + x - 5$

Following is a statement of a theorem about certain cubic equations. For this theorem, $b$ represents a real number.

• Theorem A If $f$ is a cubic function of the form $f(x) = x^3 - x + b$ and $b > 1$, then the function $f$ has exactly one $x$-intercept.

Following is another theorem about $x$-intercepts of functions:

• Theorem B If $f$ and $g$ are functions with $g(x) = k \cdot f(x)$, where $k$ is a nonzero real number, then $f$ and $g$ have exactly the same $x$-intercepts.

Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas?

$k(x) = 2x^3 + 2x + 3$

Following is a statement of a theorem about certain cubic equations. For this theorem, $b$ represents a real number.

• Theorem A If $f$ is a cubic function of the form $f(x) = x^3 - x + b$ and $b > 1$, then the function $f$ has exactly one $x$-intercept.

Following is another theorem about $x$-intercepts of functions:

• Theorem B If $f$ and $g$ are functions with $g(x) = k \cdot f(x)$, where $k$ is a nonzero real number, then $f$ and $g$ have exactly the same $x$-intercepts.

Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas?

$r(x) = x^4 - x + 11$

Following is a statement of a theorem about certain cubic equations. For this theorem, $b$ represents a real number.

• Theorem A If $f$ is a cubic function of the form $f(x) = x^3 - x + b$ and $b > 1$, then the function $f$ has exactly one $x$-intercept.

Following is another theorem about $x$-intercepts of functions:

• Theorem B If $f$ and $g$ are functions with $g(x) = k \cdot f(x)$, where $k$ is a nonzero real number, then $f$ and $g$ have exactly the same $x$-intercepts.

Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas?

$F(x) = 2x^3 - 2x + 7$

Using what you learned in this section, answer the following.

Is the set of natural numbers closed under division?

Using what you learned in this section, answer the following.

Is the set of rational numbers closed under division?

Using what you learned in this section, answer the following.

Is the set of nonzero rational numbers closed under division?

Using what you learned in this section, answer the following.

Is the set of positive rational numbers closed under division?

Using what you learned in this section, answer the following.

Is the set of positive real numbers closed under subtraction?

Using what you learned in this section, answer the following.

Is the set of negative rational numbers closed under division?

Using what you learned in this section, answer the following.

Is the set of negative integers closed under addition?