Inverse Functions: Show (a) Analytically & (b) Graphically

In summary, an inverse function is a function that performs the opposite operation of another function. To find the inverse of a function, you can use the steps of rewriting the function, swapping the variables, solving for y, and replacing y with f<sup>-1</sup>(x). It is important to find the inverse of a function for solving equations, understanding relationships between variables, and determining properties of the function. The inverse of a function can be shown analytically using algebraic methods or the horizontal line test, and graphically by plotting the original and inverse functions on the same coordinate plane.
  • #1
duki
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Homework Statement



Show that f and g are inverse functions (a) analytically and (b) graphically.

f(x) = 5x+1
g(x) = (x-1)/5

Homework Equations



I've got (a), but I'm unsure at how to solve for (b).

The Attempt at a Solution



Here's my (a): [tex]f(g(x)) = 5(x-1/5) + 1 = x[/tex]
How do I solve graphically?
 
Last edited:
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  • #2
The inverse of a function is the function reflected across the line x=y.
 

1) What is an inverse function?

An inverse function is a function that performs the opposite operation of another function. In other words, if a function f(x) takes an input x and produces an output y, then its inverse function, denoted as f-1(y), takes an input y and produces an output x.

2) How do you find the inverse of a function?

To find the inverse of a function, you can use the following steps:1. Rewrite the function in the form y = f(x).2. Swap the x and y variables.3. Solve for y.4. Replace y with f-1(x) to get the inverse function.

3) Why is it important to find the inverse of a function?

Finding the inverse of a function can be useful in solving certain types of equations and in understanding the relationship between two variables. It can also help in finding the domain and range of a function, and in determining whether a function is one-to-one or onto.

4) How do you show the inverse of a function analytically?

To show the inverse of a function analytically, you can use the algebraic method described in the second question. You can also use the horizontal line test, which states that a function and its inverse are inverse functions if and only if a horizontal line intersects their graphs at most once.

5) How do you show the inverse of a function graphically?

To show the inverse of a function graphically, you can plot the original function and its inverse on the same coordinate plane. If the two graphs are reflections of each other across the line y = x, then they are inverse functions. You can also use a graphing calculator to graph the inverse function.

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