1. Problem: Given $f(x) = 2 - x$ and $g(x) = -2x + 3$, find:\ a. $f + 2g$ and evaluate $(f + 2g)(2)$\ b. $2f - g$ and evaluate $(2f - g)(2)$\ c. $2f \, . \, g$ and evaluate $(2f \, . \, g)(2)$\ d. $\frac{3f}{2g}$ and evaluate $\left(\frac{3f}{2g}\right)(2)$ Step 1: Calculate $f + 2g$\ $f(x) + 2g(x) = (2 - x) + 2(-2x + 3) = 2 - x - 4x + 6 = 8 - 5x$ Step 2: Evaluate $(f + 2g)(2)$\ $8 - 5(2) = 8 - 10 = -2$ Step 3: Calculate $2f - g$\ $2f(x) - g(x) = 2(2 - x) - (-2x + 3) = 4 - 2x + 2x - 3 = 1$ Step 4: Evaluate $(2f - g)(2)$\ Since $2f - g = 1$, $(2f - g)(2) = 1$ Step 5: Calculate $2f \, . \, g$ (product)\ $2f(x) \, . \, g(x) = 2(2 - x)(-2x + 3) = 2( -4x + 6 + 2x^2 - 3x) = 2(2x^2 - 7x + 6) = 4x^2 - 14x + 12$ Step 6: Evaluate $(2f \, . \, g)(2)$\ $4(2)^2 - 14(2) + 12 = 4(4) - 28 + 12 = 16 - 28 + 12 = 0$ Step 7: Calculate $\frac{3f}{2g}$\ $\frac{3f(x)}{2g(x)} = \frac{3(2 - x)}{2(-2x + 3)} = \frac{6 - 3x}{-4x + 6}$ Step 8: Evaluate $\left(\frac{3f}{2g}\right)(2)$\ $\frac{6 - 3(2)}{-4(2) + 6} = \frac{6 - 6}{-8 + 6} = \frac{0}{-2} = 0$ 2. Problem: Find domain and range of relation $R$ where $(x, y)$ are natural numbers satisfying $2x + y = 10$ Step 1: Domain are natural numbers $x$ such that $2x + y = 10$ has a natural number $y$ Step 2: Express $y$ in terms of $x$: $y = 10 - 2x$ Step 3: Since $x, y \in \mathbb{N}$, $y > 0$, so $10 - 2x > 0 \Rightarrow 2x < 10 \Rightarrow x < 5$ Step 4: The natural $x$ values are $1, 2, 3, 4$ Step 5: Corresponding $y$ values: $x=1 \Rightarrow y=8$, $x=2 \Rightarrow y=6$, $x=3 \Rightarrow y=4$, $x=4 \Rightarrow y=2$ Step 6: Domain = ${1, 2, 3, 4}$, Range = ${2, 4, 6, 8}$ 3. Problem: Find x- and y-intercepts and sketch graph of:\ a. $x + y = 2$\ b. $y = -3x - 5$ Step 1 (a): For $x + y = 2$, x-intercept is where $y=0$: $x + 0 = 2 \Rightarrow x=2$ Step 2 (a): y-intercept is where $x=0$: $0 + y = 2 \Rightarrow y=2$ Step 3 (b): For $y = -3x - 5$, x-intercept is where $y=0$: $0 = -3x - 5 \Rightarrow 3x = -5 \Rightarrow x = -\frac{5}{3}$ Step 4 (b): y-intercept is where $x=0$: $y = -3(0) - 5 = -5$ Summary: - Line 1 intercepts: x=2, y=2 - Line 2 intercepts: x= -5/3, y= -5 These intercepts help sketch the lines on a Cartesian plane with typical bottom-left axes. Final Answers: - (f + 2g)(2) = $-2$ - (2f - g)(2) = $1$ - (2f . g)(2) = $0$ - (3f / 2g)(2) = $0$ - Domain of R = ${1,2,3,4}$, Range of R = ${2,4,6,8}$ - Line $x + y=2$ intercepts: $(2,0)$ and $(0,2)$ - Line $y = -3x - 5$ intercepts: $(-\frac{5}{3},0)$ and $(0,-5)$