Review (MATH 8) I.MC. * Rtors \( (1-3) \) : Let \( f(x)=\sqrt{x-3}, g(x)=2 x-3 \) 1) Find \( [g \circ f](7) \). 2) \( \left(\frac{g}{t}\right)(12) \) 3.) \( (f+9)(4) \) 4.) Fud the dmanan o the netatinn \( 2(x+3)=18 \). 3.) If \( f(x)=\sqrt{x+2} \), anthet \( x \) the durian of \( f \). 4.) \( F f(x)=3 x-2 \). Find \( f(t) \). 7.) IF \( f(x)=3 x-2 \). Find the value \( 4 x \) which \( f(x)=4 \). * Reters \( (8-10) \) : Let \( f(x)=x^{2}-1, g(x)=\frac{2}{x+1}, h(x)=2 x \) 8.) Find \( \left[h \circ \frac{f}{g}\right](1) \) 9.) Determine the domain of \( (g \cdot h)(x) \), iv.) Deternmle the doman of \( \left(\frac{h}{f}\right)(x) \). II. IDENTIF C GTION: A. Deteronie the standard forms of the FF Functions. 1.) Constand function: 4.) Identity funation: 3.) Qualuatic function: 4) Saume mod function: -5) Aludate Value function:

Sure, I'll help you work through the problems you've listed. Let's tackle them step-by-step. --- ### **I. Multiple Choice Problems** **Operators (1–10):** Given: - \( f(x) = \sqrt{x - 3} \) - \( g(x) = 2x - 3 \) - Later functions: - \( f(x) = x^2 - 1 \) - \( g(x) = \frac{2}{x + 1} \) - \( h(x) = 2x \) --- #### **1) Find \( [g \circ f](7) \)** **Solution:** \[ [g \circ f](7) = g(f(7)) \] First, find \( f(7) \): \[ f(7) = \sqrt{7 - 3} = \sqrt{4} = 2 \] Then, find \( g(2) \): \[ g(2) = 2(2) - 3 = 4 - 3 = 1 \] **Answer:** \( [g \circ f](7) = 1 \) --- #### **2) Find \( \left(\frac{g}{f}\right)(12) \)** **Solution:** \[ \left(\frac{g}{f}\right)(12) = \frac{g(12)}{f(12)} \] Calculate \( g(12) \): \[ g(12) = 2(12) - 3 = 24 - 3 = 21 \] Calculate \( f(12) \): \[ f(12) = \sqrt{12 - 3} = \sqrt{9} = 3 \] \[ \left(\frac{g}{f}\right)(12) = \frac{21}{3} = 7 \] **Answer:** \( \left(\frac{g}{f}\right)(12) = 7 \) --- #### **3) Find \( (f + 9)(4) \)** **Solution:** Assuming the problem is \( (f + 9)(4) = f(4) + 9 \): \[ f(4) = \sqrt{4 - 3} = \sqrt{1} = 1 \] \[ (f + 9)(4) = 1 + 9 = 10 \] **Answer:** \( (f + 9)(4) = 10 \) --- #### **4) Solve the equation \( 2(x + 3) = 18 \)** **Solution:** \[ 2(x + 3) = 18 \] Divide both sides by 2: \[ x + 3 = 9 \] Subtract 3 from both sides: \[ x = 6 \] **Answer:** \( x = 6 \) --- #### **5) If \( f(x) = \sqrt{x + 2} \), find the domain of \( f \).** **Solution:** The expression inside the square root must be non-negative: \[ x + 2 \geq 0 \] \[ x \geq -2 \] **Answer:** The domain of \( f \) is all real numbers \( x \) such that \( x \geq -2 \). --- #### **6) If \( F f(x) = 3x - 2 \), find \( f(t) \).** **Solution:** Assuming \( F \) is a notation error and \( f(x) = 3x - 2 \): \[ f(t) = 3t - 2 \] **Answer:** \( f(t) = 3t - 2 \) --- #### **7) If \( f(x) = 3x - 2 \), find the value of \( x \) for which \( f(x) = 4 \).** **Solution:** \[ 3x - 2 = 4 \] Add 2 to both sides: \[ 3x = 6 \] Divide by 3: \[ x = 2 \] **Answer:** \( x = 2 \) --- #### **8) Find \( \left[h \circ \left(\frac{f}{g}\right)\right](1) \)** Given: - \( f(x) = x^2 - 1 \) - \( g(x) = \frac{2}{x + 1} \) - \( h(x) = 2x \) **Solution:** \[ \left(\frac{f}{g}\right)(1) = \frac{f(1)}{g(1)} \] Calculate \( f(1) \): \[ f(1) = 1^2 - 1 = 0 \] Calculate \( g(1) \): \[ g(1) = \frac{2}{1 + 1} = \frac{2}{2} = 1 \] \[ \left(\frac{f}{g}\right)(1) = \frac{0}{1} = 0 \] Now, apply \( h \): \[ h(0) = 2(0) = 0 \] **Answer:** \( \left[h \circ \left(\frac{f}{g}\right)\right](1) = 0 \) --- #### **9) Determine the domain of \( (g \cdot h)(x) \)** Given: - \( g(x) = \frac{2}{x + 1} \) - \( h(x) = 2x \) **Solution:** \[ (g \cdot h)(x) = \frac{2}{x + 1} \cdot 2x = \frac{4x}{x + 1} \] The denominator cannot be zero: \[ x + 1 \neq 0 \] \[ x \neq -1 \] **Answer:** The domain is all real numbers except \( x = -1 \). --- #### **10) Determine the domain of \( \left(\frac{h}{f}\right)(x) \)** Given: - \( h(x) = 2x \) - \( f(x) = x^2 - 1 \) **Solution:** \[ \left(\frac{h}{f}\right)(x) = \frac{2x}{x^2 - 1} \] The denominator cannot be zero: \[ x^2 - 1 \neq 0 \] \[ x \neq 1 \] \[ x \neq -1 \] **Answer:** The domain is all real numbers except \( x = 1 \) and \( x = -1 \). --- ### **II. Identification of Standard Function Forms** **A. Determine the standard forms of the following functions:** 1. **Constant Function:** \[ f(x) = c \] - *Description:* A function that always returns the same constant value regardless of the input \( x \). 2. **Identity Function:** \[ f(x) = x \] - *Description:* A function that returns the input value as the output. 3. **Quadratic Function:** \[ f(x) = ax^2 + bx + c \] - *Description:* A polynomial function of degree 2, where \( a \neq 0 \). 4. **Absolute Value Function:** \[ f(x) = |x| \] - *Description:* A function that returns the non-negative value of \( x \). 5. **Additional Functions (Assumed from Context):** - **Sum Function:** \[ f(x) = x + k \] - *Description:* A linear function representing the addition of a constant \( k \) to the input \( x \). - **Modulus Function:** Likely referring to periodic functions like sine or cosine, but based on the list, it seems to correspond to the absolute value function already listed. - **Arithmetic Operations on Functions:** Functions involving addition, subtraction, multiplication, or division of other functions. **Note:** The list provided had some typographical errors, so the above interpretations are based on standard function forms commonly studied in mathematics. --- If you have specific questions about any of these problems or need further clarification on any topic, feel free to ask!