Respuesta :
Answer: The inverse of the linear function f(x)=2x+1 is f^(-1) (x) = (1/2)x-1/2
Solution
f(x)=2x+1
y=f(x)
y=2x+1
Isolating x: Subtracting 1 both sides of the equation:
y-1=2x+1-1
y-1=2x
Multiplying both sides of the equation by 1/2:
(1/2)(y-1)=(1/2)2x
(1/2)y-1/2=x
x=(1/2)y-1/2
Changing "x" by "f^(-1) (x)" and "y" by "x":
f^(-1) (x) = (1/2)x-1/2
The inverse of the function f(x) = 2x + 1 is [tex]f^{-1} (x) = \frac{x-1}{2}[/tex]
From the question the given function is
f(x) = 2x +1
To determine the inverse of this function,
First, let y = f(x)
That is,
y = 2x +1
Then, make x the subject of the equation
y = 2x + 1
To make x the subject of this equation,
First, subtract 1 from both sides
We get
[tex]y - 1 = 2x +1 -1[/tex]
[tex]y - 1 = 2x[/tex]
∴ [tex]2x = y -1[/tex]
Now, divide both sides by 2
[tex]\frac{2x}{2} =\frac{y-1}{2}[/tex]
∴ [tex]x =\frac{y-1}{2}[/tex]
Now, rewrite as f⁻¹(x) by replacing y by x
That is,
[tex]f^{-1} (x) = \frac{x-1}{2}[/tex]
Hence, the inverse of the function f(x) = 2x + 1 is [tex]f^{-1} (x) = \frac{x-1}{2}[/tex]
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