The inverse of a function is an important concept in mathematics. It refers to a function that reverses the effect of another function. In other words, if you apply a function and then apply its inverse, you get back to where you started. In this blog post, we will discuss the inverse of f(x) = 2x – 10.

To find the inverse of a function, we need to switch x and y in the equation and solve for y. So, let’s do that for f(x) = 2x – 10:

x = 2y – 10

x + 10 = 2y

y = (x + 10) / 2

Therefore, the inverse of f(x) = 2x – 10 is:

f^-1(x) = (x + 10) / 2

Now that we have the inverse, let’s discuss some key properties.

Firstly, the domain and range of the inverse are switched compared to the original function. The domain of f(x) is all real numbers, but the domain of f^-1(x) is also all real numbers. The range of f(x) is all real numbers less than or equal to -10 (since the line has a slope of 2, meaning it’s decreasing), but the range of f^-1(x) is all real numbers.

Secondly, the composition of a function and its inverse always results in the identity function. That is, if we apply f(x) and then f^-1(x), we get back to x. Let’s check that:

f(f^-1(x)) = f((x + 10) / 2)

= 2((x + 10) / 2) – 10

= x + 10 – 10

= x

Therefore, f(f^-1(x)) = x, which is the identity function.

Similarly, if we apply f^-1(x) and then f(x), we also get back to x:

f^-1(f(x)) = f^-1(2x – 10)

= (2x – 10 + 10) / 2

= x/2

But we want to find f(f^-1(x)), so let’s substitute x/2 in for x:

f(f^-1(x)) = 2((x/2)) – 10

= x – 10

Therefore, f^-1(f(x)) = x – 10, which is also the identity function.

Finally, let’s graph f(x) and f^-1(x) on the same coordinate plane.

As we can see from the graph, the inverse of f(x) reflects the graph over the line y = x. This makes sense because the inverse function undoes what the original function does. So, if we apply f(x) to a point on the graph of f^-1(x), we end up on the graph of f(x). If we then apply f^-1(x), we undo what f(x) did and end up back where we started on the graph of f^-1(x).

In conclusion, the inverse of f(x) = 2x – 10 is f^-1(x) = (x + 10) / 2. The domain and range of the inverse are switched compared to the original function, and the composition of a function and its inverse always results in the identity function. We can also graph f(x) and f^-1(x) on the same coordinate plane to see how they relate to each other.