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1: Numbers and Functions
https://math.libretexts.org/Bookshelves/Precalculus/APEX_PreCalculus_(Chapman_Herald_and_Libertini)/01%3A_Numbers_and_Functions
In calculus, you will be manipulating functions to answer application questions such as optimizing the volume of a soda can while minimizing the material used to make it or computing the volume and ma...In calculus, you will be manipulating functions to answer application questions such as optimizing the volume of a soda can while minimizing the material used to make it or computing the volume and mass of a small caliber projectile from an engineering drawing. Logarithmic functions are used in many measurement scales such as the Richter scale that measures the strength of an earthquake and are even used to measure the loudness of sound in decibels.
2.4: Graphs and Graphing
https://math.libretexts.org/Bookshelves/Precalculus/APEX_PreCalculus_(Chapman_Herald_and_Libertini)/02%3A_Basic_Skills_for_Calculus/2.04%3A_Graphs_and_Graphing
For example, if the slope is $-\frac{2}{3}$ , you would start at the initial point, move $3$ units to the right (because the horizontal change is 3), and then move down $2$ units in the y-directi...For example, if the slope is $-\frac{2}{3}$ , you would start at the initial point, move $3$ units to the right (because the horizontal change is 3), and then move down $2$ units in the y-direction (because the slope is negative and the vertical change is 2). (Note: you can also move vertically and then horizontally, either way gives the same result.) Plot this new point, and then use a straightedge to connect both points.
1.3: Factoring and Expanding
https://math.libretexts.org/Bookshelves/Precalculus/APEX_PreCalculus_(Chapman_Herald_and_Libertini)/01%3A_Numbers_and_Functions/1.03%3A_Factoring_and_Expanding
Instead, we like to think about starting with the first term in the first set of parentheses and multiplying it by the first term of the second set, then the second term of the second set, then the th...Instead, we like to think about starting with the first term in the first set of parentheses and multiplying it by the first term of the second set, then the second term of the second set, then the third term of the second set, etc. We know that if x is positive, $x^3$ and $x^2$ are also positive, and we can’t add up a bunch of positive numbers and get 0, so we don’t need to check any of the positive factors, only the negative factors.
1.1: Real Numbers
https://math.libretexts.org/Bookshelves/Precalculus/APEX_PreCalculus_(Chapman_Herald_and_Libertini)/01%3A_Numbers_and_Functions/1.01%3A_Real_Numbers
When working with units, you first need to make sure all units are consistent; for example, if you are finding the area of a square and one side is measured in feet and the other side in inches, you w...When working with units, you first need to make sure all units are consistent; for example, if you are finding the area of a square and one side is measured in feet and the other side in inches, you will need to convert so that both sides have the same units.
1.4.1: Exercises 1.4
https://math.libretexts.org/Bookshelves/Precalculus/APEX_PreCalculus_(Chapman_Herald_and_Libertini)/01%3A_Numbers_and_Functions/1.04%3A_Radicals_and_Exponents/1.4.01%3A_Exercises_1.4
$\displaystyle \Bigg( \sqrt{x} + \frac{1}{\sqrt{x}} \Bigg)^2$ $\displaystyle (\sqrt{x})^2 + \Bigg(\frac{1}{\sqrt{x}} \Bigg)^2$ $\displaystyle \Bigg(\sqrt[3]{x} +1 \Bigg)^3$ \(\displaystyle \Bigg... $\displaystyle \Bigg( \sqrt{x} + \frac{1}{\sqrt{x}} \Bigg)^2$ $\displaystyle (\sqrt{x})^2 + \Bigg(\frac{1}{\sqrt{x}} \Bigg)^2$ $\displaystyle \Bigg(\sqrt[3]{x} +1 \Bigg)^3$ $\displaystyle \Bigg( \frac{-5x^{-1/4}y^3}{x^{1/4}y^{1/2}}\Bigg)^2$ $\displaystyle \Bigg( \frac{-2x^{2/3}y^2}{x^{-2}y^{1/2}}\Bigg)^6$ $\displaystyle \frac{(e^{x+3})^2}{e^{-x}}$ $\displaystyle \sqrt[3]{\Bigg(\frac{e^{4\theta-6}y^2}{e^{\theta}y^{-4}}\Bigg)}$
2.2.1: Exercises 2.2
https://math.libretexts.org/Bookshelves/Precalculus/APEX_PreCalculus_(Chapman_Herald_and_Libertini)/02%3A_Basic_Skills_for_Calculus/2.02%3A_Solving_Inequalities/2.2.01%3A_Exercises_2.2
No, the statement $x^2 >0$ is always true, but has a break point at $x=0$ . $x \leq -5$ and $x>2$ $3 \leq x <4$ or $x>4$ $5 < x \leq 6$ or $7 \leq x <8$ In exercises $\PageIndex{15}$ ...No, the statement $x^2 >0$ is always true, but has a break point at $x=0$ . $x \leq -5$ and $x>2$ $3 \leq x <4$ or $x>4$ $5 < x \leq 6$ or $7 \leq x <8$ In exercises $\PageIndex{15}$ - $\PageIndex{26}$ , solve the given inequality and express your answer in interval notation. $\displaystyle x \in (-\infty,-5) \cup (3,\infty)$ $\displaystyle x \in (-\infty,2) \cup [7, \infty)$ $\displaystyle x \in (-\infty,-4)\cup [-2,-1] \cup (4,\infty)$
2.3.1: Exercises 2.3
https://math.libretexts.org/Bookshelves/Precalculus/APEX_PreCalculus_(Chapman_Herald_and_Libertini)/02%3A_Basic_Skills_for_Calculus/2.03%3A_Function_Domains/2.3.01%3A_Exercises_2.3
What does it mean if $x=2$ is in the domain of $f(x)$ ? What does it mean if $x=4$ is not in the domain of $f(x)$ ? False; it depends on both the domain of $f(x)$ and the domain of $g(x)$ . I...What does it mean if $x=2$ is in the domain of $f(x)$ ? What does it mean if $x=4$ is not in the domain of $f(x)$ ? False; it depends on both the domain of $f(x)$ and the domain of $g(x)$ . In exercises $\PageIndex{5}$ - $\PageIndex{7}$ , express the domain of the given function using interval notation. $x \in (-\infty,-2)\cup (4,\infty)$ In exercises $\PageIndex{12}$ - $\PageIndex{22}$ , express the domain of the given function using interval notation.
1.5: Logarithms and Exponential Functions
https://math.libretexts.org/Bookshelves/Precalculus/APEX_PreCalculus_(Chapman_Herald_and_Libertini)/01%3A_Numbers_and_Functions/1.05%3A_Logarithms_and_Exponential_Functions
For this statement, we are answering the question “ $\frac{1}{9}$ is 3 raised to what power?” We saw in the previous question that $\log_3{(9)} = 2$ which gives us a hint that our answer is related...For this statement, we are answering the question “ $\frac{1}{9}$ is 3 raised to what power?” We saw in the previous question that $\log_3{(9)} = 2$ which gives us a hint that our answer is related to 2, but that we need $3^2$ to be in the denominator.
1.2.1: Exercises 1.2
https://math.libretexts.org/Bookshelves/Precalculus/APEX_PreCalculus_(Chapman_Herald_and_Libertini)/01%3A_Numbers_and_Functions/1.02%3A_Introduction_to_Functions/1.2.01%3A_Exercises_1.2
Each of exercise $\PageIndex{5}$ – $\PageIndex{8}$ is some combination of $f(x)$ , $g(x)$ , and $h(x)$ . For example, $x^{3} + x + 4$ is the addition of $f(x)$ and $g(x)$ and can be writt...Each of exercise $\PageIndex{5}$ – $\PageIndex{8}$ is some combination of $f(x)$ , $g(x)$ , and $h(x)$ . For example, $x^{3} + x + 4$ is the addition of $f(x)$ and $g(x)$ and can be written as $f(x) + g(x)$ . In exercises $\PageIndex{9}$ – $\PageIndex{11}$ , determine the input variable of each function, any parameters of the function, and the type of function.
1.3.1: Exercises 1.3
https://math.libretexts.org/Bookshelves/Precalculus/APEX_PreCalculus_(Chapman_Herald_and_Libertini)/01%3A_Numbers_and_Functions/1.03%3A_Factoring_and_Expanding/1.3.01%3A_Exercises_1.3
$f(x) = x(x+ \frac{1-\sqrt{5}}{2})(x+\frac{1+\sqrt{5}}{2})$ ; the factor $x$ pairs with the root $x=0$ , the factor $x + \frac{1-\sqrt{5}}{2}$ pairs with the root $x=\frac{-1+\sqrt{5}}{2}$ , an... $f(x) = x(x+ \frac{1-\sqrt{5}}{2})(x+\frac{1+\sqrt{5}}{2})$ ; the factor $x$ pairs with the root $x=0$ , the factor $x + \frac{1-\sqrt{5}}{2}$ pairs with the root $x=\frac{-1+\sqrt{5}}{2}$ , and the factor $x+\frac{1-\sqrt{5}}{2}$ pairs with the root $x=\frac{-1-\sqrt{5}}{2}$
2.1.1: Exercises 2.1
https://math.libretexts.org/Bookshelves/Precalculus/APEX_PreCalculus_(Chapman_Herald_and_Libertini)/02%3A_Basic_Skills_for_Calculus/2.01%3A_Linear_Functions/2.1.01%3A_Exercises_2.1
Is this the y-intercept of the line or the x-intercept of the line? Write the equation in point-slope form for the line that passes through $(1,2)$ and is parallel to the line $2x+y=5$ . Write the ...Is this the y-intercept of the line or the x-intercept of the line? Write the equation in point-slope form for the line that passes through $(1,2)$ and is parallel to the line $2x+y=5$ . Write the equation in point-slope form for the line that passes through $(0,4)$ and is perpendicular to the line $x-2y=6$ . Write the equation in slope-intercept form for the line that passes through $(-2,2)$ and is perpendicular to the line $x+3y=8$ .

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