Question

How do I find the inverse of the first function ?

Answer 1

`\boxed{\boxed{\pink{\sf \leadsto Hence \ the \ Inverse \ of \ the \ given \ function \ is \ x^2+2x-4 .}}}`

Explanation:

A function is given to us , and we need to find its inverse. So the function is ,

`\bf\implies f(x) = √(x + 5 ) - 1`

So , firstly replace x with y in the given function and then solve for y to get its inverse . Taking the given function ,

`\bf \implies f(x) = √(x+5)-1\\\\\bf\implies y = √(x+5)-1 \\\\\bf\implies x = √(y+5)-1 \:\:\bigg\lgroup \red{\sf Replacing \ x \ with \ y .} \bigg\rgroup \\\\\bf\implies x + 1 = √(y+5)\\\\\bf\implies y+5 = (x+1)^2 \\\\\bf\implies y+5 = x^2+1 +2x \\\\\bf\implies y = x^2+1-5+2x \\\\\bf\implies y = x^2-4+2x \\\\\bf\implies \boxed{\red{\bf f^(-1)(x)= x^2+2x - 4 }}`

Here I have also attached the graph of , x² + 2x - 4 and √(x+5) - 1. Here blue graph is of √(x+5) - 1 & red of x²+2x-4.

Hence the inverse f -¹(x) of the given function is x² + 2x - 4 .