Find Range Algebraically

[nycmath]

See attachment.

[Eigenvalue]

1. Let R(x)=y and solve for x, in terms of y, to find the range
y=[tex]\frac{1}{x-1}[/tex]+1
y=[tex]\frac{x}{x-1}[/tex] rewrite 1 as [tex]\frac{x-1}{x-1}[/tex]
y(x-1)=x
yx-y=x
-y=x-yx
-y=x(1-y)
2. Find the values of y that are undefined
x=[tex]\frac{-y}{1-y}[/tex]

The range excludes the value y=1, as the denominator equals 0 for y=1; a fraction's denominator cannot be 0 because dividing by 0 is undefined

(-infinity, 1) U (1, infinity)

[nycmath]

1. Let R(x)=y and solve for x, in terms of y, to find the range
y=[tex]\frac{1}{x-1}[/tex]+1
y=[tex]\frac{x}{x-1}[/tex] rewrite 1 as [tex]\frac{x-1}{x-1}[/tex]
y(x-1)=x
yx-y=x
-y=x-yx
-y=x(1-y)
2. Find the values of y that are undefined
x=[tex]\frac{-y}{1-y}[/tex]

The range excludes the value y=1, as the denominator equals 0 for y=1; a fraction's denominator cannot be 0 because dividing by 0 is undefined

(-infinity, 1) U (1, infinity)

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