Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
3. Functions
Function Operations
Problem 115Lial - 13th Edition
Textbook Question
Let ƒ(x) = √(x-2) and g(x) = x^2. Find each of the following, if possible. (g ○ ƒ)(3)
![](/channels/images/assetPage/verifiedSolution.png)
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Step 1: Understand the composition of functions. The notation means , which involves applying the function first and then applying the function to the result.
Step 2: Start by finding . Substitute into the function .
Step 3: Calculate . Simplify the expression inside the square root.
Step 4: Use the result from as the input for . Substitute this result into .
Step 5: Calculate by squaring the result from . This gives you .
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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. The notation (g ○ ƒ)(x) means to apply function ƒ first and then apply function g to the result. In this case, you would first evaluate ƒ(3) and then use that output as the input for g.
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Domain of a Function
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For ƒ(x) = √(x-2), the expression under the square root must be non-negative, meaning x must be greater than or equal to 2. Understanding the domain is crucial for determining valid inputs when composing functions.
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Domain Restrictions of Composed Functions
Evaluating Functions
Evaluating a function involves substituting a specific value into the function's equation to find the corresponding output. For example, to find (g ○ ƒ)(3), you first evaluate ƒ(3) and then substitute that result into g. This process requires careful calculation and an understanding of how to manipulate algebraic expressions.
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Related Practice
Textbook Question
Without using paper and pencil, evaluate each expression given the following functions. ƒ(x)=x+1 and g(x)=x^2
(ƒ+g)(2)
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