Answer:
[tex]f(-1) = 8[/tex]
[tex]f^{-1}(x) = \frac{5}{3} - \frac{x}{3}[/tex]
Step-by-step explanation:
Given
[tex]f(x) = 5 - 3x[/tex]
Required
[tex]f(-1)[/tex] and [tex]f^{-1}(x)[/tex]
Solving for f(-1)
Substitute -1 for x in [tex]f(x) = 5 - 3x[/tex]
[tex]f(-1) = 5 - 3(-1)[/tex]
[tex]f(-1) = 5 + 3[/tex]
[tex]f(-1) = 8[/tex]
Solving for [tex]f^{-1}(x)[/tex]
Let y = f(x)
[tex]y = 5 - 3x[/tex]
Interchange the position of x and y
[tex]x = 5 - 3y[/tex]
Make y the subject of formula (add 3y to both sides)
[tex]3y + x = 5 - 3y + 3y[/tex]
[tex]3y + x = 5[/tex]
Subtract x from both sides
[tex]3y + x - x = 5 - x[/tex]
[tex]3y = 5 - x[/tex]
Divide through by 3
[tex]\frac{3y}{3} = \frac{5}{3} - \frac{x}{3}[/tex]
[tex]y = \frac{5}{3} - \frac{x}{3}[/tex]
Replace y with [tex]f^{-1}(x)[/tex]
[tex]f^{-1}(x) = \frac{5}{3} - \frac{x}{3}[/tex]
