Problem 54
Question
Determine whether the function is one-to-one. If it is, find its inverse function. \(f(x)=a x+b, \quad a \neq 0\)
Step-by-Step Solution
Verified Answer
The function \(f(x) = ax + b\) is one-to-one and its inverse function is \(f^{-1}(x) = a^{-1}(x - b)\)
1Step 1: Check If Function is One-to-One
We start with the function \(f(x) = ax + b\). For this function, if you pick any two different x values, their corresponding y values will also be different because of the fact that \(a \neq 0\). This means the function is one-to-one.
2Step 2: Find the Inverse Function
To find the inverse of the function \(f(x)=ax+b\), we interchange the x and y variables. This gives us the equation \(x=ay+b\). We then solve this equation for y, isolating y on one side to obtain the inverse function. When we solve for y, we get \(y = a^-1 * (x - b)\). This is the inverse function, denoted as \(f^{-1}(x)\).
Key Concepts
Determining one-to-one functionsFinding inverse functionsLinear functions
Determining one-to-one functions
When exploring functions, a foundational concept is determining whether a function is one-to-one. A function is considered one-to-one if each input (x-value) produces a unique output (y-value). This uniqueness criterion means that no two different inputs yield the same output. To test whether the linear function
\(f(x) = ax + b\), where
\(a eq 0\), is one-to-one, we examine the dependency of output on input. Since the coefficient \(a\) is not zero, each input is scaled uniquely before the constant \(b\) is added, guaranteeing that every output will be distinct for different inputs. Thus, proving the function's one-to-one nature. This property is essential for the existence of an inverse function, ensuring that for every output there's exactly one corresponding input.
\(f(x) = ax + b\), where
\(a eq 0\), is one-to-one, we examine the dependency of output on input. Since the coefficient \(a\) is not zero, each input is scaled uniquely before the constant \(b\) is added, guaranteeing that every output will be distinct for different inputs. Thus, proving the function's one-to-one nature. This property is essential for the existence of an inverse function, ensuring that for every output there's exactly one corresponding input.
Finding inverse functions
An inverse function essentially reverses the operation of the given function, turning outputs back into their original inputs. Finding the inverse of a one-to-one function starts by swapping the roles of the independent variable \(x\) and the dependent variable \(y\). For the linear equation
\(f(x) = ax + b\), we restate it as
\(x = ay + b\). The next step involves solving this equation for \(y\), isolating it to express the inverse function, \(f^{-1}(x)\). By subtracting \(b\) from both sides and then multiplying by the reciprocal of \(a\), \(a^{-1}\), we obtain the inverse function:
\(y = a^{-1} * (x - b)\). The ability to find inverse functions is crucial as it provides a method to 'undo' the original function, offering insights into function symmetry and the connections between inputs and outputs.
\(f(x) = ax + b\), we restate it as
\(x = ay + b\). The next step involves solving this equation for \(y\), isolating it to express the inverse function, \(f^{-1}(x)\). By subtracting \(b\) from both sides and then multiplying by the reciprocal of \(a\), \(a^{-1}\), we obtain the inverse function:
\(y = a^{-1} * (x - b)\). The ability to find inverse functions is crucial as it provides a method to 'undo' the original function, offering insights into function symmetry and the connections between inputs and outputs.
Linear functions
Linear functions are the simplest type of algebraic function and are represented by the formula
\(f(x) = ax + b\), where \(a\) and \(b\) are constants and \(a eq 0\). The graph of a linear function is always a straight line. The coefficient \(a\) is referred to as the function's slope, dictating its steepness and direction, while \(b\), the y-intercept, represents the point where the line crosses the y-axis.
A key property of linear functions is that they are always one-to-one unless the slope \(a\) is zero, which would make the function horizontal and not one-to-one. This distinct characteristic of having exactly one output for each input makes linear functions particularly straightforward to work with concerning inverses and function operations.
\(f(x) = ax + b\), where \(a\) and \(b\) are constants and \(a eq 0\). The graph of a linear function is always a straight line. The coefficient \(a\) is referred to as the function's slope, dictating its steepness and direction, while \(b\), the y-intercept, represents the point where the line crosses the y-axis.
A key property of linear functions is that they are always one-to-one unless the slope \(a\) is zero, which would make the function horizontal and not one-to-one. This distinct characteristic of having exactly one output for each input makes linear functions particularly straightforward to work with concerning inverses and function operations.
Other exercises in this chapter
Problem 54
In Exercises 41–64, find the derivative of the function. $$ h(t)=\frac{\ln t}{t} $$
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Evaluating a Definite Integral In Exercises \(49-56\) , evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{0}^{1} \frac{x-1}
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Find the derivative of the function. \(y=8 \arcsin \frac{x}{4}-\frac{x \sqrt{16-x^{2}}}{2}\)
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In Exercises 55–60, evaluate the integral. $$ \int_{0}^{\ln 2} \tanh x d x $$
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