Points found on y = h(x) are (7, -6) and (-2,-1).
Using these two points, we will solve for the exact equation of y = h(x).
To solve the equation, we will get the slope (m) of the two points first using the following formula:
[tex]m=\frac{y_2-y_1}{x_2-x_1}=\frac{-1-(-6)}{-2-7}=\frac{5}{-9}=-\frac{5}{9}[/tex]Now that we have a slope, we can now proceed in solving the equation using Point-Slope Formula.
[tex]\begin{gathered} y-y_1=m(x-x_1) \\ y-(-6)=-\frac{5}{9}(x-7) \\ y+6=-\frac{5}{9}(x-7) \\ 9y+54=-5x+35 \\ 9y=-5x+35-54 \\ 9y=-5x-19 \\ y=-\frac{5}{9}x-\frac{19}{9} \end{gathered}[/tex]Now that we have the equation of the dashed line, we will now solve for its inverse function y = h^-1 (x).
To solve for the inverse, we will reverse y and x with each other. The new equation will be:
[tex]x=-\frac{5}{9}y-\frac{19}{9}[/tex]From that equation, we will now equate or isolate y.
[tex]\begin{gathered} x=-\frac{5}{9}y-\frac{19}{9} \\ x=-\frac{5y-19}{9} \\ 9x=-5y-19 \\ 5y=-9x-19 \\ y=-\frac{9}{5}x-\frac{19}{5} \end{gathered}[/tex]In this equation, our slope (m) here is -9/5 and our y-intercept is at (0, -19/5). The graph for this equation will look like this.
Drag the endpoints of the solid segment to the coordinates shown above to graph y = h^-1 (x).
Or drag the endpoints to (-6,7) and (-1,-2). It's the same graph anyway.
