MATH SOLVE

4 months ago

Q:
# Which graph represents the function f(x)=3x-2/x-2

Accepted Solution

A:

ANSWER

The last graph is the correct answer

EXPLANATION

The function,

[tex]y = \frac{3x - 2}{x - 2} [/tex] is a rational function.

This rational function has a vertical asymptote

at where the denominator is zero.

That is,

[tex]x - 2 = 0[/tex]

This means that, the vertical asymptote occurs at

[tex]x = 2[/tex]

The graph also has a horizontal asymptote at,

[tex]y = \frac{3}{1} [/tex]

Thus, the horizontal asymptote occurs at

[tex]y = 3[/tex]

At x-intercept,

[tex]f(x) = 0[/tex]

This implies that,

[tex] \frac{3x - 2}{x - 2} = 0[/tex]

[tex]\Rightarrow \: 3x - 2 = 0[/tex]

[tex]3x = 2[/tex]

[tex]\Rightarrow \: x = \frac{2}{3} [/tex]

The graph cuts the x-axis at,

[tex]( \frac{2}{3} ,0)[/tex]

At y-intercept,

[tex]x = 0[/tex]

This implies that,

[tex]f(0) = \frac{3(0) - 2}{0 - 2} = \frac{ - 2}{ - 2} = 1[/tex]

The graph cuts the y-axis at,

[tex](0,1)[/tex]

The graph that satisfy all the above conditions is the last one.

The last graph is the correct answer

EXPLANATION

The function,

[tex]y = \frac{3x - 2}{x - 2} [/tex] is a rational function.

This rational function has a vertical asymptote

at where the denominator is zero.

That is,

[tex]x - 2 = 0[/tex]

This means that, the vertical asymptote occurs at

[tex]x = 2[/tex]

The graph also has a horizontal asymptote at,

[tex]y = \frac{3}{1} [/tex]

Thus, the horizontal asymptote occurs at

[tex]y = 3[/tex]

At x-intercept,

[tex]f(x) = 0[/tex]

This implies that,

[tex] \frac{3x - 2}{x - 2} = 0[/tex]

[tex]\Rightarrow \: 3x - 2 = 0[/tex]

[tex]3x = 2[/tex]

[tex]\Rightarrow \: x = \frac{2}{3} [/tex]

The graph cuts the x-axis at,

[tex]( \frac{2}{3} ,0)[/tex]

At y-intercept,

[tex]x = 0[/tex]

This implies that,

[tex]f(0) = \frac{3(0) - 2}{0 - 2} = \frac{ - 2}{ - 2} = 1[/tex]

The graph cuts the y-axis at,

[tex](0,1)[/tex]

The graph that satisfy all the above conditions is the last one.