Problem 39
Question
In Exercises \(17-40,\) let \(f(x)=-x^{2}+x, g(x)=\frac{2}{x+1},\) and \(h(x)=-2 x+1 .\) Evaluate each of the following. $$\left(\frac{f}{h}\right)(1)$$
Step-by-Step Solution
Verified Answer
The value of \(\left(\frac{f}{h}\right)(1)\) is \(0\)
1Step 1: Find \(f(1)\)
Substitute \(x=1\) in \(f(x)\) to find the value of \(f(1)\). Therefore, \(f(1)=-1^{2}+1=-1+1=0\)
2Step 2: Find \(h(1)\)
Substitute \(x=1\) in \(h(x)\) to find the value of \(h(1)\). Therefore, \(h(1)=-2*1+1=-2+1=-1\)
3Step 3: Calculate \(\left(\frac{f}{h}\right)(1)\)
The value of the fraction \(\left(\frac{f}{h}\right)(1)\) is found by substituting \(f(1)\) and \(h(1)\) into the fraction. Therefore, \(\left(\frac{f}{h}\right)(1)= \frac{f(1)}{h(1)} = \frac{0}{-1} = 0\)
Key Concepts
Evaluating FunctionsFunction OperationsComposite Functions
Evaluating Functions
When we talk about evaluating functions in precalculus, we're referring to the process of finding the value of a function for a specific input. In the given exercise, the function f(x) is defined as f(x) = -x^2 + x, and you're asked to evaluate f(1). This simply means finding the value of the function when x is 1.
Here's how you perform such an evaluation step by step: You start by replacing every instance of x in the function's formula with the given input, in this case, 1. Then you simplify using arithmetic operations. For f(x), this becomes f(1) = -(1)^2 + 1, which simplifies to 0. In the same manner, evaluating other functions such as g(x) and h(x) with specific inputs follows the same routine and is crucial for more complex operations involving functions.
Here's how you perform such an evaluation step by step: You start by replacing every instance of x in the function's formula with the given input, in this case, 1. Then you simplify using arithmetic operations. For f(x), this becomes f(1) = -(1)^2 + 1, which simplifies to 0. In the same manner, evaluating other functions such as g(x) and h(x) with specific inputs follows the same routine and is crucial for more complex operations involving functions.
Function Operations
Precalculus involves various types of function operations, including addition, subtraction, multiplication, division, and composition of functions. These operations follow the same arithmetic principles as with numbers, but instead, you perform them on functions. In the exercise, you encounter the division of two functions, specifically f(x) divided by h(x).
To divide one function by another – denoted as \(\frac{f}{h}\) – you first evaluate both functions separately at the same input value (as done in the previous explanation). After finding f(1) and h(1), you would divide these two resulting numbers just as you would with any regular division of numbers. Thus, \(\frac{f(1)}{h(1)}\) results in \(\frac{0}{-1}\), which simplifies to 0. It's essential to understand that function operations can lead to undefined values if you divide by a function value that is zero, but in this case, it works out neatly.
To divide one function by another – denoted as \(\frac{f}{h}\) – you first evaluate both functions separately at the same input value (as done in the previous explanation). After finding f(1) and h(1), you would divide these two resulting numbers just as you would with any regular division of numbers. Thus, \(\frac{f(1)}{h(1)}\) results in \(\frac{0}{-1}\), which simplifies to 0. It's essential to understand that function operations can lead to undefined values if you divide by a function value that is zero, but in this case, it works out neatly.
Composite Functions
A composite function is formed when one function is applied to the result of another function. The notation (f ∘ g)(x), or f(g(x)), signifies that the function g is evaluated first, and its result is then used as the input for the function f. In more complex scenarios, you might encounter several functions composed together, such as f(g(h(x))), and these can be evaluated step by step as well.
However, the exercise provided doesn't deal with a composite function but rather a simple division of functions. Understanding composite functions is valuable when you need to evaluate expressions where one function effectively 'feeds' into another. To understand this concept deeply, practice evaluating composite functions step by step, and remember to start with the innermost function progressing to the outer.
However, the exercise provided doesn't deal with a composite function but rather a simple division of functions. Understanding composite functions is valuable when you need to evaluate expressions where one function effectively 'feeds' into another. To understand this concept deeply, practice evaluating composite functions step by step, and remember to start with the innermost function progressing to the outer.
Other exercises in this chapter
Problem 39
Find \(x+y, x-y, x y,\) and \(x / y\). $$x=-3+5 i ; y=2-3 i$$
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Use the verbal description to find an algebraic expression for the function. The graph of the function \(g(t)\) is formed by translating the graph of \(f(t)=|t|
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Solve the quadratic equation by completing the square. $$2 x^{2}+8 x-1=0$$
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Decide if each function is odd, even, or neither by using the definitions. $$f(x)=2 x$$
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