Question

Can someone help me on How to solve inverse functions ?Find the inverse of f(x)=x^2-5x+1

Answer 1

To find the inverse function of f(x) = x^2 - 5x + 1, one should first replace f(x) with y, swap x and y, and then solve the quadratic equation for y. The inverse function may have two branches and the original function might need domain restriction to ensure the inverse is also a function.

Step-by-step explanation:

Finding the Inverse Function

To find the inverse of the function f(x) = x^2 - 5x + 1, we need to follow a sequence of steps. The relationship between a function and its inverse basically undoes the original function. For example, the exponential e^x is undone by the natural log ln(x). Similarly, to find the inverse of a quadratic function, we have to express y in terms of x and then solve for x. The process involves several steps:

1. Replace f(x) with y: y = x^2 - 5x + 1.
2. Swap the roles of x and y: x = y^2 - 5y + 1.
3. Solve the resulting quadratic equation for y. This can be challenging as the equation may not factor easily and might require the quadratic formula for solution.
4. The solutions for y give us the two branches of the inverse function, which correspond to the positive and negative square roots, if applicable.

However, it's worth noting that not all functions have inverses that are also functions, especially when dealing with quadratics which are not one-to-one unless their domain is restricted. In the case of x^2 - 5x + 1, the function is not one-to-one over all real numbers, because it does not pass the Horizontal Line Test. Therefore, we would usually restrict the domain to find an inverse that is also a function. For example, we might only consider x >= 5/2, where 5/2 is the vertex of the parabola described by the quadratic function.

Answer 2

The given function is:

`f(x)=x^2-5x+1`

To find the inverse, first you need to check whether the function is bijective, which means:

1) The function should be one-one.

2) The function shoudl be onto.

Graph the function as shown below:

As seen from the graph, the green curve represents the function.

Now draw horizontal and vertical lines on the function.

If the horizontal lines pass through more than one points which can be clearly seen from the graph, the function is not one-one.

Hence the function is not bijective and therefore the inverse of such function does not exist.