Use the table to evaluate the given compositions. <IMAGE>


ƒ(ƒ(h(3)))

Verified step by step guidance

1. Identify the innermost function in the composition, which is \( h(3) \).
2. Use the table to find the value of \( h(3) \).
3. Substitute the value of \( h(3) \) into the next function in the composition, \( f(h(3)) \).
4. Use the table to find the value of \( f(h(3)) \).
5. Substitute the value of \( f(h(3)) \) into the outermost function, \( f(f(h(3))) \), and use the table to find the final value.

Key Concepts

Function Composition

Function composition involves applying one function to the result of another function. If you have two functions, f(x) and g(x), the composition f(g(x)) means you first apply g to x, and then apply f to the result. This concept is fundamental in calculus as it allows for the evaluation of complex expressions by breaking them down into simpler parts.

Evaluating Functions

Evaluating a function means substituting a specific input value into the function to find the output. For example, if f(x) = x + 2, then f(3) = 3 + 2 = 5. In the context of function composition, you will need to evaluate the inner function first before using its output as the input for the outer function.

Nested Functions

Nested functions occur when a function is applied within another function. In the expression f(f(h(3))), h(3) is evaluated first, then the result is used as the input for f, and finally, that output is again used as the input for f. Understanding how to handle nested functions is crucial for correctly evaluating compositions in calculus.