## Inverse functions word problems

## Word problem on inverse variation

Let f be a one-one onto function from point A to point B. Assume that y is an arbitrary B element. Then, since f is on, there must be an element x in A such that f(x) = y. As a consequence, we can define a function f-1 asf-1 : B —> A: f-1(y) = xif and only if f(x) = y. The inverse of f is the above function f-1. If and only if f is one-one onto, a function is invertible. To put it another way, the one-one onto function has an inverse function. For any function f, you might be able to find f-1. But only if f is a one-one onto function can f-1 be a function. If f is one-one onto, then f-1 is one-one onto as well. For example, if f = A —> B, then f-1 = B—-> A.

Let’s assume f(x) = x + k. (k is a constant).

Step 1: Substitute f(x) with y in the previous function.

Then we get y = x + ky = x + k, where “y” has defined “x” in terms of “y.” Step 2: We must now redefine y = x + k in terms of “x” in terms of “y.” Then we’ll have x = y – k. Step 3: Substitute “x” with f-1 (x) and “y” with “x” in x = y – k. As a result, the inverse of f(x) is f-1(x) = x – k.

## Inverse function problems

You appear to be using a computer with a “small” screen (i.e. you are probably on a mobile phone). This site is best viewed in landscape mode due to the simplicity of the mathematics. Many of the calculations will run off the side of your device if it is not in landscape mode (you should be able to navigate to see them) and some of the menu items will be cut off due to the small screen width.

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[requirecolor beginalign*fleft( colorProcessBlue – 1 right) & = 3left( – 1 right)] (requirecolor) (requirecolor) (requirecolor) (requirecolor) (requirecolor) (requirecolor) (requirecolor) (requirecolor) (requirecolor) (requirecolor) (requirecolor) (requirecolor) (requirecolor) ( – 2 = colorRed – 5 & hspace0.25in hspace0.25in hspace0.25in hspace0.25in hspace0.25in hspace0.25in hspace0.25in hspace0.25in hspace hspace0.25ingleft( colorRed – 5 right)rightarrow hspace0.25ingleft( colorRed – 5 right)rightarrow hspace0.25ingleft( colorRed – 5 right)rightarrow hspace0.25ingleft( color & = frac – 53% + frac23% = frac – 33% = colorProcessBlue – 1 & & gleft( colorProcessBlue 2 right) & = frac23% + frac23% = colorRedfrac43% & hspace0.25in Rightarrow hspace0.25infleft( colorRed frac43 right) & = 3left( frac43 right) hspace0.25infleft( colorRed frac43 right) hspace0.25infleft( colorRed frac43 right) hspace0.25infleft( frac43 right) hspace0.25infleft( frac43 right) hspace0.25infleft( frac43 right) colorProcessBlue 2 endalign*] – 2 = 4 – 2 = colorProcessBlue 2 endalign*]

In the first example, we got a value of -5 by plugging (x = – 1) into (fleft( x right)). We then turned around and plugged (x = – 5) into (gleft( x right)) to get a value of -1, which was the original number.

### Solving problems involving inverse function

When we use function notation to describe a relationship, we know that for a given value of x, there will only be one possible value of y. We can deduce from the equation f(x) = 2x + 1 that f(5) = 11 and that no other value is possible. As a consequence, when x equals 5, we can assume that the function f equals 11.

We may also say “reverse” when we find the inverse of a function. However, we refer to this as an opposite rather than a reverse. The inverse function f-1(x) = (x – 1)/2 looks like this. The inverse function f-1 equals 5 when x is 11.

By flipping the variables x and y, the inverse can be found. Furthermore, the range of the original function becomes the domain of the inverse, and the domain of the original function becomes the range of the inverse. We’ll get some practice with these definitions using multiple representations in the Multiple representations of functions and inverses exercise before we get into the algebra of finding the inverse. The equations will be given by your teacher.

### Inverse function from word

When we solve for x as the dependent variable and y as the independent variable and rename them according to normal convention, we get the inverse of a function. If f(x) represents the number of bacteria in a culture as a function of time (time is the independent variable, and bacteria is the dependent variable), then f -1(x) represents the time it takes for the number of bacteria in the culture to reach a certain value (bacteria is now the independent variable, and time is the dependent variable). The inverse function is written “f inverse” and is denoted as f -1(x). In terms of graphics, it’s the inverse of the given function along the line y = x.

A function must be one-to-one in order to have an inverse. This simply means that f can never have the same value twice (i.e., no two values yield the same f(x) value). A function is one-to-one if and only if a horizontal line cannot cross its graph more than once, according to the horizontal line test.

The inverse function has a domain equal to the original function’s range and a range equal to the original function’s domain; thus, if a function has a domain of A and a range of B, the inverse function has a domain of B and a range of A. This makes sense since an inverse can be thought of as ‘flipping’ the function along the line y = x, exchanging both the x and y values.