# How do you find the inverse of #f(x)=x^2+2x-5# and is it a function?

Let us recall that, the inverse of a given function exists, if and only if , the given function is a bijection , i.e., the given function is a 1 - 1 ( one-to-one , or, injection ) and onto ( or, surjection ).

By signing up, you agree to our Terms of Service and Privacy Policy

To find the inverse of ( f(x) = x^2 + 2x - 5 ), follow these steps:

- Replace ( f(x) ) with ( y ).
- Swap the variables ( x ) and ( y ) to form an equation in terms of ( x ) and ( y ).
- Solve the equation obtained in step 2 for ( y ), which will give you the expression for the inverse function ( f^{-1}(x) ).
- Replace ( y ) with ( f^{-1}(x) ) to represent the inverse function.

Step 1: Replace ( f(x) ) with ( y ): ( y = x^2 + 2x - 5 )

Step 2: Swap the variables ( x ) and ( y ): ( x = y^2 + 2y - 5 )

Step 3: Solve for ( y ): ( x = y^2 + 2y - 5 ) ( 0 = y^2 + 2y - x - 5 )

Using the quadratic formula, ( y = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} ), where ( a = 1 ), ( b = 2 ), and ( c = -x - 5 ), we get:

( y = \frac{{-2 \pm \sqrt{{2^2 - 4(1)(-x - 5)}}}}{{2(1)}} ) ( y = \frac{{-2 \pm \sqrt{{4 + 4x + 20}}}}{{2}} ) ( y = \frac{{-2 \pm \sqrt{{4x + 24}}}}{{2}} ) ( y = \frac{{-2 \pm \sqrt{{4(x + 6)}}}}{{2}} ) ( y = \frac{{-2 \pm 2\sqrt{{x + 6}}}}{{2}} ) ( y = -1 \pm \sqrt{{x + 6}} )

Step 4: Replace ( y ) with ( f^{-1}(x) ): ( f^{-1}(x) = -1 \pm \sqrt{{x + 6}} )

Therefore, the inverse function of ( f(x) = x^2 + 2x - 5 ) is ( f^{-1}(x) = -1 \pm \sqrt{{x + 6}} ).

As for whether the inverse function is a function itself, it is not a function because it does not pass the vertical line test. For any given value of ( x ), there are two possible corresponding values of ( y ). Hence, it is not a one-to-one function and thus not a function.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you express the following function, f(x) as a composition of two functions f and g given #f(x)=x^2/(x^2+4)#?
- How do you find the domain of #f(x)=|3x-2|#?
- Given #-f(x)#, how do you describe the transformation?
- In order for the function #f(x)=k(x-1)+x(k+3)+2# to be a constant function, what should be the value of #k# ?
- How do you find the Vertical, Horizontal, and Oblique Asymptote given #y = (8 x^2 + x - 2)/(x^2 + x - 72)#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7