Question

The function y = f (x) is linear.

Answer 1

Answer:

$f(x)=2x-6$

$f^{-1}(x)=\dfrac{x+6}{2}$

Step-by-step explanation:

Using slope-intercept form of linear function:  $y = mx + b$

$\implies f(x)=mx+b$

$\textsf{if} \ \ f(1) = -4$

$\implies m + b=-4$

$\implies b = -4-m$

Find inverse of slope-intercept form:

swap x and y:  $x = my + b$

Make y the subject:

$\implies x - b = my$

$\implies y = \dfrac{x-b}{m}$

$\implies f^{-1}(x) = \dfrac{x-b}{m}$

$\textsf{if} \ \ f^{-1}(0) = 3$

$\implies \dfrac{0-b}{m}=3$

$\implies b=-3m$

$\textsf{equation 1:} \ \ b = -4-m$

$\textsf{equation 2:} \ \ b=-3m$

Equate the equations and solve for $m$:

$\implies b=b$

$\implies -4-m=-3m$

$\implies -4=-2m$

$\implies m=2$

Substitute found value for $m$ into one of the equations and solve for $b$:

$b = -4-2=-6$

Substitute found values of $m$ and $b$ into equations for $f(x)$ and $f^{-1}(x)$:

$f(x)=2x-6$

$f^{-1}(x)=\dfrac{x+6}{2}$