Answer to Question #243512 in Algebra for Trixie Saavedra

Question #243512

Find the domain and range of the following rational function. Use any notation. 1. f(x) = 2

π‘₯+1

2. f(x) = 3π‘₯ π‘₯+3

3. f(x) = 3βˆ’π‘₯ π‘₯βˆ’7

4. f(x) = 2+π‘₯ π‘₯

5. f(x) = (π‘₯+1) π‘₯2βˆ’1


1
Expert's answer
2021-09-29T07:24:35-0400

1. f(x)=2x+1f(x)=\dfrac{2}{x+1}

x+1=ΜΈ0=>x=ΜΈβˆ’1x+1\not=0=>x\not=-1


Domain:(βˆ’βˆž,βˆ’1)βˆͺ(βˆ’1,∞)Domain:(-\infin, -1)\cup (-1, \infin)

Range:(βˆ’βˆž,0)βˆͺ(0,∞)Range:(-\infin, 0)\cup (0, \infin)

2. f(x)=3xx+3f(x)=\dfrac{3x}{x+3}

x+3=ΜΈ0=>x=ΜΈβˆ’3x+3\not=0=>x\not=-3


f(x)=3xx+3=3x+9βˆ’9x+3=3βˆ’9x+3f(x)=\dfrac{3x}{x+3}=\dfrac{3x+9-9}{x+3}=3-\dfrac{9}{x+3}


Domain:(βˆ’βˆž,βˆ’3)βˆͺ(βˆ’3,∞)Domain:(-\infin, -3)\cup (-3, \infin)

Range:(βˆ’βˆž,3)βˆͺ(3,∞)Range:(-\infin, 3)\cup (3, \infin)


3. f(x)=3βˆ’xxβˆ’7f(x)=\dfrac{3-x}{x-7}

xβˆ’7=ΜΈ0=>x=ΜΈ7x-7\not=0=>x\not=7


f(x)=3βˆ’xxβˆ’7=3βˆ’(xβˆ’7)βˆ’7xβˆ’7=βˆ’1βˆ’4xβˆ’7f(x)=\dfrac{3-x}{x-7}=\dfrac{3-(x-7)-7}{x-7}=-1-\dfrac{4}{x-7}


Domain:(βˆ’βˆž,7)βˆͺ(7,∞)Domain:(-\infin, 7)\cup (7, \infin)

Range:(βˆ’βˆž,βˆ’1)βˆͺ(βˆ’1,∞)Range:(-\infin, -1)\cup (-1, \infin)


4. f(x)=2+xxf(x)=\dfrac{2+x}{x}

x=ΜΈ0x\not=0

f(x)=2+xx=1+2xf(x)=\dfrac{2+x}{x}=1+\dfrac{2}{x}Domain:(βˆ’βˆž,0)βˆͺ(0,∞)Domain:(-\infin, 0)\cup (0, \infin)

Range:(βˆ’βˆž,1)βˆͺ(1,∞)Range:(-\infin, 1)\cup (1, \infin)



5. f(x)=x+1x2βˆ’1f(x)=\dfrac{x+1}{x^2-1}

x2βˆ’1=ΜΈ0=>x=ΜΈβˆ’1,x=ΜΈ1x^2-1\not=0=>x\not=-1, x\not=1


f(x)=x+1x2βˆ’1=x+1(x+1)(xβˆ’1)=βˆ’1xβˆ’1,x=ΜΈΒ±1f(x)=\dfrac{x+1}{x^2-1}=\dfrac{x+1}{(x+1)(x-1)}=-\dfrac{1}{x-1}, x\not=\pm1


Domain:(βˆ’βˆž,βˆ’1)βˆͺ(βˆ’1,1)βˆͺ(1,∞)Domain:(-\infin,-1)\cup (-1,1)\cup (1, \infin)

Range:(βˆ’βˆž,0)βˆͺ(0,∞)Range:(-\infin, 0)\cup (0, \infin)


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