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| { | ||
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| "# Algebra Introduction\n", | ||
| "\n", | ||
| "This section introduces the basic concepts of algebra, including variables, constants, and functions\n", | ||
| "\n", | ||
| "## Functions\n", | ||
| "A function is a rule that takes one or more inputs and produces a single output. For example, the function $f(x) = x + 1$ takes a single input $x$, adds one to it, and produces a single output.\n", | ||
| "In algebra, functions are written using symbols and formulas. For example, the function $f(x) = x + 1$ can be written as $f:x \\rightarrow x + 1$.\n", | ||
| "The input to a function is called the **argument** or **input variable**. The output is called the **value** or **output variable**.\n", | ||
| "\n", | ||
| "Functions are often written using the following notation:\n", | ||
| "\n", | ||
| "$$y = f(x)$$\n", | ||
| "\n", | ||
| "The notation above is read as \"$y$ equals $f$ of $x$\" or \"$y$ is a function of $x$\".\n", | ||
| "The notation above is useful because it allows us to define a function without specifying its name. For example, we can define a function $f$ as follows:\n", | ||
| "\n", | ||
| "$$f(x) = x^2$$\n", | ||
| "\n", | ||
| "We can then use the function $f$ to compute the square of any number. For example, $f(2) = 2^2 = 4$ and $f(3) = 3^2 = 9$.\n", | ||
| "\n", | ||
| "$$\n", | ||
| "\\mathrm{f}(x) = \\sqrt{x + {6}} \\\\\n", | ||
| "\\mathrm{f}(6) = \\sqrt{10 + {6}} \\\\\n", | ||
| "\\mathrm{f}(6) = 4.0\n", | ||
| "$$\n", | ||
| "\n", | ||
| "$$\n", | ||
| "\\begin{gathered}\n", | ||
| "f(x)=\\frac{x-3}{x+2} \\\\\n", | ||
| "f(3)=\\frac{3-3}{3+2}=\\frac{0}{5}=0\n", | ||
| "\\end{gathered}\n", | ||
| "$$\n", | ||
| "\n", | ||
| "### Domain and Range of a Function\n", | ||
| "The **domain** of a function is the set of all possible inputs to the function. The **range** of a function is the set of all possible outputs of the function.\n", | ||
| "For example, the function $f(x) = x^2$ has a domain of all real numbers and a range of all non-negative real numbers.\n", | ||
| "The domain of a function is often written as $D(f)$ and the range is often written as $R(f)$.\n", | ||
| "\n", | ||
| "### Piecewise Functions\n", | ||
| "A piecewise function is a function that is defined by multiple sub-functions, each sub-function applying to a different interval of the main function's domain.\n", | ||
| "For example, the function $f(x) = |x|$ is defined by two sub-functions:\n", | ||
| "\n", | ||
| "$$\n", | ||
| "f(x) = \\begin{cases}\n", | ||
| "x & \\text{if } x \\geq 0 \\\\\n", | ||
| "-x & \\text{if } x < 0\n", | ||
| "\\end{cases}\n", | ||
| "\n", | ||
| "$$\n", | ||
| "\n", | ||
| "\n", | ||
| "## Expoents\n", | ||
| "An exponent is a number that indicates how many times a base number is multiplied by itself.\n", | ||
| "For example, $2^3$ is the same as $2 \\times 2 \\times 2$ and $2^4$ is the same as $2 \\times 2 \\times 2 \\times 2$.\n", | ||
| "The number $2$ is called the **base** and the number $3$ is called the **exponent**.\n", | ||
| "Exponents are often written using the following notation:\n", | ||
| "\n", | ||
| "$$2^3 = 2 \\times 2 \\times 2 = 8$$\n", | ||
| "\n", | ||
| "The notation above is read as \"two to the power of three\" or \"two cubed\".\n", | ||
| "\n", | ||
| "### Negative Exponents\n", | ||
| "A negative exponent indicates that the base number should be divided by itself a certain number of times.\n", | ||
| "For example, $2^{-3}$ is the same as $\\frac{1}{2^3}$ and $2^{-4}$ is the same as $\\frac{1}{2^4}$.\n", | ||
| "The number $2$ is called the **base** and the number $-3$ is called the **exponent**.\n", | ||
| "Negative exponents are often written using the following notation:\n", | ||
| "\n", | ||
| "$$2^{-3} = \\frac{1}{2^3} = \\frac{1}{8}$$\n", | ||
| "\n", | ||
| "The notation above is read as \"two to the power of negative three\" or \"two to the power of minus three\".\n", | ||
| "### Fractional Exponents\n", | ||
| "A fractional exponent indicates that the base number should be multiplied by itself a certain number of times.\n", | ||
| "For example, $2^{\\frac{1}{2}}$ is the same as $\\sqrt{2}$ and $2^{\\frac{1}{3}}$ is the same as $\\sqrt[3]{2}$.\n", | ||
| "The number $2$ is called the **base** and the number $\\frac{1}{2}$ is called the **exponent**.\n", | ||
| "Fractional exponents are often written using the following notation:\n", | ||
| "\n", | ||
| "$$2^{\\frac{1}{2}} = \\sqrt{2} = 1.414213562373095$$\n", | ||
| "\n", | ||
| "The notation above is read as \"two to the power of one half\" or \"two to the power of one over two\".\n", | ||
| "\n", | ||
| "## Logarithms\n", | ||
| "A logarithm is the inverse of an exponent. For example, the logarithm of $2^3$ is $3$.\n", | ||
| "The logarithm of a number $x$ to the base $b$ is written as $\\log_b(x)$.\n", | ||
| "For example, $\\log_2(8) = 3$ because $2^3 = 8$.\n", | ||
| "\n", | ||
| "### Common Logarithms\n", | ||
| "\n", | ||
| "The common logarithm of a number $x$ is the logarithm of $x$ to the base $10$. The common logarithm of $x$ is written as $\\log(x)$.\n", | ||
| "For example, $\\log(100) = 2$ because $10^2 = 100$.\n", | ||
| "\n", | ||
| "### Natural Logarithms\n", | ||
| "The natural logarithm of a number $x$ is the logarithm of $x$ to the base $e$. The natural logarithm of $x$ is written as $\\ln(x)$.\n", | ||
| "For example, $\\ln(100) = 4.60517$ because $e^{4.60517} = 100$.\n", | ||
| "\n", | ||
| "## Polynomials\n", | ||
| "\n", | ||
| "A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.\n", | ||
| "\n", | ||
| "For example, $x^2 + 2x + 1$ is a polynomial because it consists of the variables $x$ and the coefficients $1$ and $2$.\n", | ||
| "\n", | ||
| "The degree of a polynomial is the highest degree of its terms. For example, the polynomial $x^2 + 2x + 1$ has a degree of $2$ because its highest degree term is $x^2$.\n" | ||
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