Directions: Read the following information and answer the two items that follow: Let f(x) = x 2, g(x) = tan x and h(x) = In x.
What is [fo(fof)](2) equal to?
256
Understanding the Problem: Function Composition Evaluation
The problem asks us to evaluate a nested function composition, specifically [f o (f o f)](2), given the function \(f(x) = x^2\). We are also given \(g(x) = \tan x\) and \(h(x) = \ln x\), but these functions are not required for this particular question based on the composition specified.
Function composition, denoted by the symbol 'o', means applying one function to the result of another function. For example, \((f \circ g)(x) = f(g(x))\).
In this case, we need to evaluate \([f \circ (f \circ f)](2)\). This can be broken down step-by-step:
- First, evaluate the innermost function: \(f(2)\).
- Second, evaluate the next layer of composition: \((f \circ f)(2)\), which is \(f(f(2))\).
- Third, evaluate the outermost composition: \([f \circ (f \circ f)](2)\), which is \(f((f \circ f)(2))\).
Step-by-Step Evaluation of \[f o (fof)](2)
Let's use the given function \(f(x) = x^2\) and evaluate the expression step by step.
Step 1: Evaluate f(2)
Substitute \(x=2\) into the function \(f(x) = x^2\):
\(f(2) = 2^2\)
\(f(2) = 4\)
Step 2: Evaluate (f o f)(2)
The composition \((f \circ f)(2)\) means \(f(f(2))\). We already found that \(f(2) = 4\). Now we need to find \(f\) of this result:
\((f \circ f)(2) = f(f(2))\)
\((f \circ f)(2) = f(4)\)
Now, substitute \(x=4\) into the function \(f(x) = x^2\):
\(f(4) = 4^2\)
\(f(4) = 16\)
So, \((f \circ f)(2) = 16\).
Step 3: Evaluate \[f o (fof)](2)
The composition \([f \circ (f \circ f)](2)\) means \(f((f \circ f)(2))\). We just found that \((f \circ f)(2) = 16\). Now we need to find \(f\) of this result:
\[f \circ (f \circ f)](2) = f((f \circ f)(2))\)
\[f \circ (f \circ f)](2) = f(16)\)
Finally, substitute \(x=16\) into the function \(f(x) = x^2\):
\(f(16) = 16^2\)
\(f(16) = 256\)
Therefore, \([f \circ (f \circ f)](2) = 256\).
Summary of Calculations
Here's a quick summary of the evaluation steps:
- \(f(x) = x^2\)
- \(f(2) = 2^2 = 4\)
- \((f \circ f)(2) = f(f(2)) = f(4) = 4^2 = 16\)
- \([f \circ (f \circ f)](2) = f((f \circ f)(2)) = f(16) = 16^2 = 256\)
Final Answer Determination
Based on our step-by-step evaluation, \([f \circ (f \circ f)](2)\) is equal to 256. We check this result against the provided options.
The options were:
- 2
- 8
- 16
- 256
Our calculated value, 256, matches one of the options.
| Step | Calculation | Result |
|---|---|---|
| 1 | \(f(2)\) | 4 |
| 2 | \(f(f(2)) = f(4)\) | 16 |
| 3 | \(f(f(f(2))) = f(16)\) | 256 |
Revision Table: Key Concepts for Function Composition
| Concept | Explanation | Notation |
|---|---|---|
| Function Composition | Applying one function to the results of another function. | \((f \circ g)(x) = f(g(x))\) |
| Evaluating Composition | Start with the innermost function and work outwards. | Evaluate \(g(x)\) first, then apply \(f\) to the result. For \((f \circ f)(x)\), evaluate \(f(x)\) then apply \(f\) again. |
| Order Matters | Generally, \((f \circ g)(x)\) is not the same as \((g \circ f)(x)\). | Example: if \(f(x)=x+1\), \(g(x)=x^2\), \((f \circ g)(x) = (x^2)+1\), \((g \circ f)(x) = (x+1)^2\). |
Additional Information: Domain and Range in Function Composition
While not directly relevant to this specific problem where we are evaluating at a point and the function is \(x^2\) (which has a domain of all real numbers), understanding the domain and range is crucial for function composition in general.
- For \((f \circ g)(x)\) to be defined, \(x\) must be in the domain of \(g\), AND \(g(x)\) must be in the domain of \(f\).
- The domain of \((f \circ g)(x)\) is the set of all \(x\) in the domain of \(g\) such that \(g(x)\) is in the domain of \(f\).
In our case with \(f(x)=x^2\), the domain of \(f\) is all real numbers. Any real number output from an inner \(f\) function (\(x^2\)) will be a non-negative real number, which is always within the domain of the outer \(f\) function. Thus, \((f \circ f)(x)\) and \([f \circ (f \circ f)](x)\) are defined for all real numbers \(x\).
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