1. Mathematics
  2. The function y = f (x) is linear.

The function y = f (x) is linear. Given f-¹ (0) = 3 and f (1) = - 4, find the formula of
f (x) and f-¹ (x)

sarah5707 sarah5707    2   14.02.2022 06:40    3

Answers
ronii73 ronii73  14.02.2022 06:40

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}

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