Textbook Question
For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=x^2-4x+2
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3. Functions
Function Operations
2:56 minutesProblem 65
Textbook Question
In Exercises 51–66, find a. (fog) (x)b. (go f) (x)c. (fog) (2)d. (go f) (2). f(x) = 1/x, g(x)= 1/x
Verified step by step guidance1
Step 1: Understand the composition of functions. The notation (f \circ g)(x) means f(g(x)), which is the function f applied to the result of g(x). Similarly, (g \circ f)(x) means g(f(x)).
Step 2: Find (f \circ g)(x). Substitute g(x) into f(x). Since f(x) = \frac{1}{x} and g(x) = \frac{1}{x}, we have (f \circ g)(x) = f(g(x)) = f(\frac{1}{x}) = \frac{1}{\frac{1}{x}}.
Step 3: Simplify (f \circ g)(x). The expression \frac{1}{\frac{1}{x}} simplifies to x. So, (f \circ g)(x) = x.
Step 4: Find (g \circ f)(x). Substitute f(x) into g(x). Since g(x) = \frac{1}{x} and f(x) = \frac{1}{x}, we have (g \circ f)(x) = g(f(x)) = g(\frac{1}{x}) = \frac{1}{\frac{1}{x}}.
Step 5: Simplify (g \circ f)(x). The expression \frac{1}{\frac{1}{x}} simplifies to x. So, (g \circ f)(x) = x.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Composition
Function composition involves combining two functions to create a new function. If f(x) and g(x) are two functions, the composition (fog)(x) means applying g first and then applying f to the result, expressed as f(g(x)). Understanding this concept is crucial for solving problems that require evaluating composite functions.
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Evaluating Functions
Evaluating functions means substituting a specific value into a function to find its output. For example, if f(x) = 1/x, evaluating f(2) involves substituting 2 for x, resulting in f(2) = 1/2. This skill is essential for calculating the values of composite functions at given points.
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Reciprocal Functions
Reciprocal functions are functions of the form f(x) = 1/x, which output the multiplicative inverse of the input. In this case, both f(x) and g(x) are reciprocal functions. Understanding their properties, such as asymptotes and behavior near zero, is important for analyzing the results of their compositions.
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