Given functions: f(x) = (2)/(3)x - 4 and g(x) = sqrt(2x). Calculate f(3), f(-2), g(-5), g(sqrt(8)). Also, calculate the images of 0 and 6 by the function f.

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Asked on April 11, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-04-11T12:18:17.795Z

Another day — let's solve it. Given functions: f(x) = (2)/(3)x - 4 g(x) = sqrt(2x) 1) Calculate f(3), f(-2), g(-5), g(sqrt(8)). Step 1: Calculate f(3). Substitute x=3 into f(x). f(3) = (2)/(3)(3) - 4 f(3) = 2 - 4 f(3) = -2 Step 2: Calculate f(-2). Substitute x=-2 into f(x). f(-2) = (2)/(3)(-2) - 4 f(-2) = -(4)/(3) - 4 f(-2) = -(4)/(3) - (12)/(3) f(-2) = -(16)/(3) Step 3: Calculate g(-5). Substitute x=-5 into g(x). g(-5) = sqrt(2(-5)) g(-5) = sqrt(-10) The square root of a negative number is not a real number. Step 4: Calculate g(sqrt(8)). Substitute x=sqrt(8) into g(x). g(sqrt(8)) = sqrt(28) Simplify sqrt(8) = sqrt(4 × 2) = 2sqrt(2). g(sqrt(8)) = sqrt(2(22)) g(sqrt(8)) = sqrt(42) We can write this as (4sqrt(2))^1/2 = (2^2 · 2^1/2)^1/2 = (2^2.5)^1/2 = 2^1.25. Alternatively, sqrt(42) = sqrt(4) · sqrt(2) = 2 · (2^1/2)^1/2 = 2 · 2^1/4. g(sqrt(8)) = 2[4]2 Summary for 1): f(3) = -2 f(-2) = -(16)/(3) g(-5) is not a real number. g(sqrt(8)) = 2[4]2 2) Calculate the images of 0 and 6 by the function f. Step 1: Calculate f(0). Substitute x=0 into f(x). f(0) = (2)/(3)(0) - 4 f(0) = 0 - 4 f(0) = -4 Step 2: Calculate f(6). Substitute x=6 into f(x). f(6) = (2)/(3)(6) - 4 f(6) = 4 - 4 f(6) = 0 Summary for 2): The image of 0 by f is -4. The image of 6 by f is 0. 3) Calculate the images of 0 and -sqrt(2) by the function g. Step 1: Calculate g(0). Substitute x=0 into g(x). g(0) = sqrt(2(0)) g(0) = sqrt(0) g(0) = 0 Step 2: Calculate g(-sqrt(2)). Substitute x=-sqrt(2) into g(x). g(-sqrt(2)) = sqrt(2(-2)) g(-sqrt(2)) = sqrt(-22) The square root of a negative number is not a real number. Summary for 3): The image of 0 by g is 0. The image of -sqrt(2) by g is not a real number. 4) Calculate the number which has 2 as its image by the function f. This means finding x such that f(x) = 2. (2)/(3)x - 4 = 2 Step 1: Add 4 to both sides. (2)/(3)x = 2 + 4 (2)/(3)x = 6 Step 2: Multiply both sides by (3)/(2). x = 6 × (3)/(2) x = 3 × 3 x = 9 The number is 9. 5) What is the antecedent of 5 by the function g? This means finding x such that g(x) = 5. sqrt(2x) = 5 Step 1: Square both sides of the equation. (sqrt(2x))^2 = 5^2 2x = 25 Step 2: Divide by 2. x = (25)/(2) The antecedent is (25)/(2). Send me the next one 📸

Accepted Answer · 2026-04-11T12:18:17.795Z

Another day — let's solve it. Given functions: f(x) = (2)/(3)x - 4 g(x) = sqrt(2x) 1) Calculate f(3), f(-2), g(-5), g(sqrt(8)). Step 1: Calculate f(3). Substitute x=3 into f(x). f(3) = (2)/(3)(3) - 4 f(3) = 2 - 4 f(3) = -2 Step 2: Calculate f(-2). Substitute x=-2 into f(x). f(-2) = (2)/(3)(-2) - 4 f(-2) = -(4)/(3) - 4 f(-2) = -(4)/(3) - (12)/(3) f(-2) = -(16)/(3) Step 3: Calculate g(-5). Substitute x=-5 into g(x). g(-5) = sqrt(2(-5)) g(-5) = sqrt(-10) The square root of a negative number is not a real number. Step 4: Calculate g(sqrt(8)). Substitute x=sqrt(8) into g(x). g(sqrt(8)) = sqrt(28) Simplify sqrt(8) = sqrt(4 × 2) = 2sqrt(2). g(sqrt(8)) = sqrt(2(22)) g(sqrt(8)) = sqrt(42) We can write this as (4sqrt(2))^1/2 = (2^2 · 2^1/2)^1/2 = (2^2.5)^1/2 = 2^1.25. Alternatively, sqrt(42) = sqrt(4) · sqrt(2) = 2 · (2^1/2)^1/2 = 2 · 2^1/4. g(sqrt(8)) = 2[4]2 Summary for 1): f(3) = -2 f(-2) = -(16)/(3) g(-5) is not a real number. g(sqrt(8)) = 2[4]2 2) Calculate the images of 0 and 6 by the function f. Step 1: Calculate f(0). Substitute x=0 into f(x). f(0) = (2)/(3)(0) - 4 f(0) = 0 - 4 f(0) = -4 Step 2: Calculate f(6). Substitute x=6 into f(x). f(6) = (2)/(3)(6) - 4 f(6) = 4 - 4 f(6) = 0 Summary for 2): The image of 0 by f is -4. The image of 6 by f is 0. 3) Calculate the images of 0 and -sqrt(2) by the function g. Step 1: Calculate g(0). Substitute x=0 into g(x). g(0) = sqrt(2(0)) g(0) = sqrt(0) g(0) = 0 Step 2: Calculate g(-sqrt(2)). Substitute x=-sqrt(2) into g(x). g(-sqrt(2)) = sqrt(2(-2)) g(-sqrt(2)) = sqrt(-22) The square root of a negative number is not a real number. Summary for 3): The image of 0 by g is 0. The image of -sqrt(2) by g is not a real number. 4) Calculate the number which has 2 as its image by the function f. This means finding x such that f(x) = 2. (2)/(3)x - 4 = 2 Step 1: Add 4 to both sides. (2)/(3)x = 2 + 4 (2)/(3)x = 6 Step 2: Multiply both sides by (3)/(2). x = 6 × (3)/(2) x = 3 × 3 x = 9 The number is 9. 5) What is the antecedent of 5 by the function g? This means finding x such that g(x) = 5. sqrt(2x) = 5 Step 1: Square both sides of the equation. (sqrt(2x))^2 = 5^2 2x = 25 Step 2: Divide by 2. x = (25)/(2) The antecedent is (25)/(2). Send me the next one 📸

Given functions: f(x) = (2)/(3)x - 4 and g(x) = sqrt(2x). Calculate f(3), f(-2), g(-5), g(sqrt(8)). Also, calculate the images of 0 and 6 by the function f.

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Given functions: f(x) = (2)/(3)x - 4 and g(x) = sqrt(2x). Calculate f(3), f(-2), g(-5), g(sqrt(8)). Also, calculate the images of 0 and 6 by the function f.

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