Respuesta :

ujalakhan18

Answer:

[tex]f^{-1}(x) = \frac{cos(2x+6)}{\pi }[/tex]

Step-by-step explanation:

[tex]y = \frac{1}{2} cos^{-1} (\pi x)-3[/tex]

Firstly, we've to interchange the variables.

[tex]x = \frac{1}{2} cos^{-1}(\pi y)-3[/tex]

Solving for y

[tex]x = \frac{cos^{-1} \pi y}{2} -3[/tex]

Adding 3 to both sides

[tex]x+3 = \frac{cos^{-1}(\pi y)}{2}[/tex]

Multiplying 2 to both sides

[tex]2(x+3) = cos^{-1} (\pi y)\\2x+6 = cos^{-1} (\pi y)[/tex]

Taking cosine on both sides

[tex]\pi y = cos (2x+6)[/tex]

Dividing both sides by y

[tex]y = \frac{cos(2x+6)}{\pi }[/tex]

Replace y by [tex]f^{-1}(x)[/tex]

=> [tex]f^{-1}(x) = \frac{cos(2x+6)}{\pi }[/tex]

09pqr4sT

Answer:

[tex]\large \boxed{f^{-1}(x)=\frac{cos(2x+6)}{\pi} }[/tex]

Step-by-step explanation:

[tex]\displaystyle y=\frac{cos^{-1} (\pi x)}{2} -3[/tex]

Swicth variables.

[tex]\displaystyle x=\frac{cos^{-1} (\pi y)}{2} -3[/tex]

Solve for y.

Add 3 to both sides.

[tex]\displaystyle x+3=\frac{cos^{-1} (\pi y)}{2}[/tex]

Multiply both sides by 2.

[tex]\displaystyle 2(x+3)=cos^{-1} (\pi y)[/tex]

[tex]\displaystyle 2x+6=cos^{-1} (\pi y)[/tex]

Take the cos of both sides.

[tex]\displaystyle cos(2x+6)=\pi y[/tex]

Divide both sides by [tex]\pi[/tex].

[tex]\displaystyle \frac{cos(2x+6)}{\pi} =y[/tex]

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