Problem 88
Question
Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm working with the linear function \(f(x)=3 x+5\) and \(I\) do not need to find \(f^{-1}\) in order to determine the value of \(\left(f \circ f^{-1}\right)(17)\).
Step-by-Step Solution
Verified Answer
The statement does make sense. There is no need to find \(f^{-1}\) to determine the value of \(\left(f \circ f^{-1}\right)(17)\) because it will always return 17, regardless of what \(f(x) = 3x+5\) is.
1Step 1: Understand inverse functions properties
We need to recall that when we have a function \(f\) and its inverse \(f^{-1}\), the composition \(\left(f \circ f^{-1}\right)(x)\) results in the original input \(x\). Therefore, we can say \(\left(f \circ f^{-1}\right)(x) = x\).
2Step 2: Apply the property
Applying this property to the given exercise, we can replace \(x\) by the given value 17, \(\left(f \circ f^{-1}\right)(17) = 17\)
3Step 3: Assess the claim
Thus, the statement 'I do not need to find \(f^{-1}\) in order to determine the value of \(\left(f \circ f^{-1}\right)(17)\)' makes sense because due to the properties of inverse functions, \(\left(f \circ f^{-1}\right)(x)\) should always return \(x\), no matter the specific \(f(x)\).
Key Concepts
Linear FunctionsComposition of FunctionsFunction Properties
Linear Functions
Linear functions are the simplest type of functions you will encounter in math. A linear function has a general form of \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. These functions graph as straight lines, hence the name "linear."
One of the key properties of linear functions involves their predictable change. The slope \(m\) represents the rate at which \(y\) changes per unit change in \(x\). For example, in the function \(f(x) = 3x + 5\), the slope is \(3\). This means for every 1 unit increase in \(x\), \(f(x)\) increases by 3 units.
Because of their straightforward nature, linear functions are easy to work with, and they set the foundation for understanding more complex functions. They are often used in real-world situations like calculating profit, speed, or any other quantity that changes at a constant rate.
One of the key properties of linear functions involves their predictable change. The slope \(m\) represents the rate at which \(y\) changes per unit change in \(x\). For example, in the function \(f(x) = 3x + 5\), the slope is \(3\). This means for every 1 unit increase in \(x\), \(f(x)\) increases by 3 units.
Because of their straightforward nature, linear functions are easy to work with, and they set the foundation for understanding more complex functions. They are often used in real-world situations like calculating profit, speed, or any other quantity that changes at a constant rate.
Composition of Functions
Composition of functions allows you to combine two or more functions into a single function, in a manner similar to substituting one function into another. Formally, the composition of two functions \(f\) and \(g\) is written as \((f \circ g)(x) = f(g(x))\).
This operation essentially tells you to apply the function \(g\) first, and then apply function \(f\) to the result of \(g\). This method is particularly useful in solving complex problems by breaking them into simpler stages.
When working with an inverse function, the composition becomes \(f(f^{-1}(x))\), which will yield \(x\). This is due to the inverse function property, which ensures that a function and its inverse "undo" each other's work. Thus, when you perform \(f \circ f^{-1}\), or \(f \circ f^{-1} (x)\), you get back to your original input value.
Understanding function composition is a stepping stone towards advanced mathematical concepts, such as transformations and function mappings.
This operation essentially tells you to apply the function \(g\) first, and then apply function \(f\) to the result of \(g\). This method is particularly useful in solving complex problems by breaking them into simpler stages.
When working with an inverse function, the composition becomes \(f(f^{-1}(x))\), which will yield \(x\). This is due to the inverse function property, which ensures that a function and its inverse "undo" each other's work. Thus, when you perform \(f \circ f^{-1}\), or \(f \circ f^{-1} (x)\), you get back to your original input value.
Understanding function composition is a stepping stone towards advanced mathematical concepts, such as transformations and function mappings.
Function Properties
Functions have various properties that define their behavior and characteristics. Some key properties include domain, range, and inverses.
One important property of inverse functions is that \((f \circ f^{-1})(x) = x\). This property is what allows you to find \(\left(f \circ f^{-1}\right)(17)\) without explicitly finding \(f^{-1}\). Such properties are crucial as they simplify complex calculations and enhance understanding in mathematical operations.
- Domain: This is the set of all possible input values \(x\) a function can accept. For linear functions like \(f(x) = 3x + 5\), the domain is all real numbers.
- Range: This is the set of all possible output values \(f(x)\). For linear functions, the range is also all real numbers, as they can produce any output value.
- Inverse: An inverse function essentially reverses the operation of the original function. If \(f(x)\) maps \(x\) to \(y\), \(f^{-1}(x)\) will map \(y\) back to \(x\). The notation \(f^{-1}\) denotes the inverse.
One important property of inverse functions is that \((f \circ f^{-1})(x) = x\). This property is what allows you to find \(\left(f \circ f^{-1}\right)(17)\) without explicitly finding \(f^{-1}\). Such properties are crucial as they simplify complex calculations and enhance understanding in mathematical operations.
Other exercises in this chapter
Problem 87
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