Of course! Let's tackle the problems step-by-step.
### Part (a)
For the function [tex]\( y = 3x - 7 \)[/tex]:
#### Reciprocal Function
To find its reciprocal function, we take the reciprocal of [tex]\( y \)[/tex]:
[tex]\[ \text{Reciprocal Function:} \quad \frac{1}{3x - 7} \][/tex]
#### Equation of Asymptote
Vertical asymptotes occur where the denominator of the reciprocal function is zero (since division by zero is undefined). So, we need to find the value of [tex]\( x \)[/tex] that makes [tex]\( 3x - 7 \)[/tex] equal to zero:
[tex]\[
3x - 7 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
3x = 7 \implies x = \frac{7}{3}
\][/tex]
Thus, the equation of the vertical asymptote is:
[tex]\[ \text{Vertical Asymptote:} \quad x = \frac{7}{3} \][/tex]
### Part (b)
For the function [tex]\( y = x^2 + 2x \)[/tex]:
#### Reciprocal Function
To find its reciprocal function, we take the reciprocal of [tex]\( y \)[/tex]:
[tex]\[ \text{Reciprocal Function:} \quad \frac{1}{x^2 + 2x} \][/tex]
#### Equations of Asymptotes
Vertical asymptotes occur where the denominator of the reciprocal function is zero. So, we need to find the values of [tex]\( x \)[/tex] that make [tex]\( x^2 + 2x \)[/tex] equal to zero:
[tex]\[
x^2 + 2x = 0
\][/tex]
Factoring out [tex]\( x \)[/tex]:
[tex]\[
x(x + 2) = 0
\][/tex]
Setting each factor equal to zero:
[tex]\[
x = 0 \quad \text{or} \quad x = -2
\][/tex]
Thus, the equations of the vertical asymptotes are:
[tex]\[ \text{Vertical Asymptotes:} \quad x = 0 \quad \text{and} \quad x = -2 \][/tex]
