The inverse of f(x)=√(x) is f^-1(x)=x^2, x ≥0. a) Graph both functions, and use them to explain

The inverse of f(x)=√(x) is f^-1(x)=x^2, x ≥0. a) Graph both...

The inverse of $f(x)=\sqrt{x}$ is $f^{-1}(x)=x^{2}, x \geq 0$. a) Graph both functions, and use them to explain why the restriction is necessary on the domain of the inverse function. b) Determine the equation, including any restrictions, of the inverse of each of the following functions. i) $g(x)=-\sqrt{x-5}$ ii) $h(x)=\sqrt{-x}+3$ iii) $j(x)=\sqrt{2 x-7}-6$

The inverse of $f(x)=\sqrt{x}$ is $f^{-1}(x)=x^{2}, x \geq 0$.
a) Graph both functions, and use them to explain why the restriction is necessary on the domain of the inverse function.
b) Determine the equation, including any restrictions, of the inverse of each of the following functions.
i) $g(x)=-\sqrt{x-5}$
ii) $h(x)=\sqrt{-x}+3$
iii) $j(x)=\sqrt{2 x-7}-6$

Key Concepts

Inverse Functions

An inverse function essentially reverses the operation of the original function, meaning that if f is a function with an inverse f?¹, then f(f?¹(x)) = x and f?¹(f(x)) = x. This concept is centrally important in understanding how outputs of a function can be mapped back to their original inputs, emphasizing the idea of reversing processes within mathematical systems.

Domain and Range Restrictions

Restrictions on the domain and range of a function are crucial when defining an inverse function, particularly when the original function is not one-to-one over its entire domain. For example, square root functions must be restricted to ensure they have a unique output for each input, which in turn allows the inverse function to be properly defined. This also ensures that the inverse function itself behaves consistently as a function.

Graphing Inverse Functions

Graphing inverse functions involves understanding that the graph of an inverse is the reflection of the original function across the line y = x. This symmetry about the line y = x provides a visual tool to verify that two functions are inverses and also illustrates why domain restrictions are necessary, as without them the reflection might not yield a one-to-one correspondence.

Finding Inverses Algebraically

Determining the inverse of a function algebraically generally involves swapping the roles of the independent and dependent variables and then solving for the new dependent variable. This process often requires careful consideration of algebraic manipulations and maintaining the necessary restrictions to ensure that the inverse function is valid over the proper domain and range.

Transcript

00:01 The concept involved in this problem is to work with inverse functions and identify the restrictions on the variable for an inverse function.

00:14 In the first part of this problem, i have graphed the function f of x equals the square root of x.

00:21 That's this black.

00:22 I'm sorry, this blue graph.

00:24 The domain of that one is the set of all of x's such as.

00:31 That x is greater than 0 being x being a real number.

00:38 And the range is a set of a y's, such that y is greater than equal to 0 with y being a real number.

00:49 Okay, then in green, i have graphed the inverse of f of x, which is x squared.

00:58 And there is a restriction placed on the variable, x is greater than equal to zero.

01:03 Now, it happens that, maybe not quite as visible in this example, but the domain of the inverse correlates with the range of the original function.

01:22 Since the inverse is formed by switching the x and y's, then you're switching the range of the original function to become the x of the inverse.

01:33 So that's why the inverse function has this restriction with it that x has to be greater than equal to zero.

01:40 Okay, so on part b of this problem, you are given some functions and you're asked to find the equation of the inverse and state any restrictions that that inverse would have.

01:55 So on the first one, to find the inverse function, i'm going to start off and i'm going to switch places with x and y.

02:05 This g of x represents the y, so i'm going to have x equals...

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