Answer:
[tex]f(x)=2x-6[/tex]
[tex]f^{-1}(x)=\dfrac{x+6}{2}[/tex]
Step-by-step explanation:
Using slope-intercept form of linear function: [tex]y = mx + b[/tex]
[tex]\implies f(x)=mx+b[/tex]
[tex]\textsf{if} \ \ f(1) = -4[/tex]
[tex]\implies m + b=-4[/tex]
[tex]\implies b = -4-m[/tex]
Find inverse of slope-intercept form:
swap x and y: [tex]x = my + b[/tex]
Make y the subject:
[tex]\implies x - b = my[/tex]
[tex]\implies y = \dfrac{x-b}{m}[/tex]
[tex]\implies f^{-1}(x) = \dfrac{x-b}{m}[/tex]
[tex]\textsf{if} \ \ f^{-1}(0) = 3[/tex]
[tex]\implies \dfrac{0-b}{m}=3[/tex]
[tex]\implies b=-3m[/tex]
[tex]\textsf{equation 1:} \ \ b = -4-m[/tex]
[tex]\textsf{equation 2:} \ \ b=-3m[/tex]
Equate the equations and solve for [tex]m[/tex]:
[tex]\implies b=b[/tex]
[tex]\implies -4-m=-3m[/tex]
[tex]\implies -4=-2m[/tex]
[tex]\implies m=2[/tex]
Substitute found value for [tex]m[/tex] into one of the equations and solve for [tex]b[/tex]:
[tex]b = -4-2=-6[/tex]
Substitute found values of [tex]m[/tex] and [tex]b[/tex] into equations for [tex]f(x)[/tex] and [tex]f^{-1}(x)[/tex]:
[tex]f(x)=2x-6[/tex]
[tex]f^{-1}(x)=\dfrac{x+6}{2}[/tex]
