Problem 31 Find $f^{-1}(x) .$ State any r... [FREE SOLUTION]

Chapter 3: Problem 31

Find $f^{-1}(x) .$ State any restrictions on the domain of $f^{-1}(x)$ $$f(x)=-2 x+5$$

Short Answer

Expert verified

The inverse function is $f^{-1}(x) = (5 - x)/2$. There are no restrictions on the domain; $f^{-1}(x)$ exists for all real numbers.

Step by step solution

Identify the Original Function

The original function is given as $f(x) = -2x + 5$. This is a linear function where -2 represents the slope and 5 is the y-intercept.

Swap $x$ and $y$

To find $f^{-1}(x)$, swap $x$ and $y$. This gives $x = -2y + 5$.

Solve for $y$

Solving for $y$ involves isolating $y$ on one side. First, subtract 5 from both sides: $x - 5 = -2y$. Then divide by -2 to get $y = -(x - 5)/2$ or $y = (5 - x)/2$. This is the equation of $f^{-1}(x)$. It's a linear function which means it exists for all real numbers. Therefore, there are no restrictions on the domain of $f^{-1}(x)$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Function

A linear function is one of the simplest types of functions. It's expressed in the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept. In our exercise, the function $f(x) = -2x + 5$ is a linear function. The slope here is $-2$, meaning for every unit increase in $x$, $f(x)$ decreases by 2 units. The y-intercept is 5, indicating that the graph crosses the y-axis at the point (0, 5). Linear functions create straight lines when graphed, making them predictable and easy to work with.

Domain Restrictions

Domain restrictions specify the set of possible input values $x$ for a function. While some functions have these restrictions (like square root or logarithmic functions), linear functions, like the one given here, generally do not. This is because the equation $-2x + 5$ can produce a corresponding output for any real number $x$. When we take the inverse $f^{-1}(x)$, it also remains linear, and thus the domain continues to be all real numbers. It's essential for students to identify when domain restrictions apply, though in this case, none are necessary.

Solving Equations

Solving equations involves finding the value of a variable that makes an equation true. For inverse functions, it often means manipulating the original equation to find $y$ in terms of $x$. In our example, to find the inverse $f^{-1}(x)$, we set $x = -2y + 5$. By rearranging this to solve for $y$, we subtract 5 to get $x - 5 = -2y$. Finally, dividing each side by -2 gives us $y = (5 - x)/2$. This process helps students understand how to transform and rearrange equations, a crucial skill in algebra.

Function Notation

Function notation is a way to express a relationship between an input variable and an output result. $f(x)$ shows that $x$ is the input to the function $f$, which results in a value $f(x)$. When finding inverse functions, notation like $f^{-1}(x)$ indicates the function that "undoes" $f(x)$. In our example, $f^{-1}(x)$ transforms an output from the original function back to its input. It's a way of denoting the reverse relationship. Understanding this notation is critical as it provides a clear, mathematical language describing these relationships and operations, aiding in grasping further mathematical concepts.

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91影视

Find \(f^{-1}(x) .\) State any restrictions on the domain of \(f^{-1}(x)\) $$f(x)=-2 x+5$$

Short Answer

Step by step solution

Identify the Original Function

Swap \(x\) and \(y\)

Solve for \(y\)

Key Concepts

Linear Function

Domain Restrictions

Solving Equations

Function Notation

One App. One Place for Learning.

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