Math Exam Solving Inequalities & Trigonometric Functions PDF

## Solving Inequalities ### Solving the Inequality $\sqrt{2x+14}-6 >4$ * **Step 1:** Add 6 to both sides $\sqrt{2x+14} > 10$ * **Step 2:** Square both sides $2x+14 > 100$ * **Step 3:** Subtract 14 from both sides $2x > 86$ * **Step 4:** Divide both sides by 2 $x > 43$ ### Solving the Inequ...

## Solving Inequalities ### Solving the Inequality $\sqrt{2x+14}-6 >4$ * **Step 1:** Add 6 to both sides $\sqrt{2x+14} > 10$ * **Step 2:** Square both sides $2x+14 > 100$ * **Step 3:** Subtract 14 from both sides $2x > 86$ * **Step 4:** Divide both sides by 2 $x > 43$ ### Solving the Inequality $\frac{2x-1}{3} > \frac{3x+1}{2}$ * **Step 1:** Multiply both sides by 6 $2(2x-1) > 3 (3x+1)$ * **Step 2:** Simplify $4x-2 > 9x+3$ * **Step 3:** Subtract 9x from both sides $-5x-2>3$ * **Step 4:** Add 2 to both sides $-5x>5$ * **Step 5:** Divide both sides by -5 (remembering to change the direction of the inequality sign because we are dividing by a negative number) $x < -1$ ### Set Notation * **Example 1:** Express the set of numbers between -1 and 5 (not including -1 and 5) using interval notation $\{x: -1 < x < 5\} = (-1, 5)$ * **Example 2:** Express the set of numbers less than or equal to -3 or greater than or equal to 3 using interval notation $\{x: x≤ -3 \text{ or } x ≥ 3\} = (-\infty, -3] U [3, \infty)$ ### Trigonometric Functions #### Cosine Function 1. The cosine function is an **even** function, meaning that $cos(-x) = cos(x)$. 2. The cosine function is a **periodic** function with period $2\pi$, meaning that $cos(x+2\pi) = cos(x)$. 3. The **domain** of the cosine function is $R$ (all real numbers). 4. The **range** of the cosine function is $[-1, 1]$ . #### Tangent Function 1. The tangent function is an **odd** function, meaning that $tan(-x) = -tan(x)$. 2. The tangent function is a **periodic** function with period $\pi$, meaning that $tan(x+\pi) = tan(x)$. 3. The **domain** of the tangent function is $R$ (all real numbers). 4. The **range** of the tangent function is $R$ (all real numbers). ## Trigonometric Identities - $sin^2(x) + cos^2(x) = 1$ - $cos(2x) = cos^2(x) - sin^2(x)$ - $cos^2(x) = \frac{1+cos(2x)}{2}$ - $sin^2(x) = \frac{1-cos(2x)}{2}$ ## **Understanding the domain and range of functions** ### Example 1: **(Image 28 - right)** * **Function:** $f(x) = (x+2)^2$ * **Domain**- The domain of a function is the set of all possible input values. In the case of f(x) = (x+2)², you can plug in any real number for x. So the domain is $x \in R = ( - \infty, \infty)$ * **Range** - The range of a function is the set of all possible output values. For f(x) = (x+2)², the smallest possible output value is 2 (when x = 0), and the function can achieve any value greater than 2. The range is therefore [2, ∞). ### Example 2: **(Image 33 - right)** * **Function:** $f(x) = \sqrt{x^2 - 4}$ * **Domain**- To find the domain, we need to consider where the function is defined. The square root of a negative number is not real. So, we need x² - 4 ≥ 0. This inequality has solutions x ≤ -2 or x ≥ 2. Hence, the domain is $x \in (-\infty,-2] U [2,\infty)$. * **Range**- As x approaches negative or positive infinity, the function f(x) approaches positive infinity. Also, the smallest possible output value is 0 (when x = 2 or x = -2). The range is therefore [0, ∞).