Without using paper and pencil, evaluate each expression given the following functions. $\text{ƒ}\left(x\right)=x+1$ and $g\left(x\right)=x^2$
(ƒg)(2)

Verified step by step guidance

1. Understand that the notation \((\f g)(2)\) means the composition of functions \(f\) and \(g\) evaluated at \(x=2\), which is \(f(g(2))\).
2. First, evaluate the inner function \(g(2)\) by substituting \$2\( into \)g(x) = x^2\(, so calculate \)g(2) = 2^2$.
3. Next, take the result from \(g(2)\) and substitute it into the function \(f(x) = x + 1\), so compute \(f(g(2)) = f(2^2) = f(4)\).
4. Evaluate \(f(4)\) by substituting \$4\( into \)f(x) = x + 1\(, which means calculating \)4 + 1$.
5. The final value of \((f g)(2)\) is the result of \(f(4)\), which completes the evaluation of the composition.

Key Concepts

Function Composition

Function composition involves applying one function to the result of another, denoted as (f∘g)(x) = f(g(x)). It means you first evaluate g at x, then use that output as the input for f. Understanding this process is essential to correctly evaluate composite functions.

Evaluating Functions

Evaluating a function means substituting a specific value for the variable and simplifying the expression. For example, if f(x) = x + 1, then f(3) = 3 + 1 = 4. This skill is necessary to find the value of functions at given points.

Order of Operations in Composite Functions

When evaluating composite functions like (f∘g)(2), you must first compute g(2) before applying f to that result. This order ensures accurate evaluation and avoids mistakes, as the inner function's output becomes the input for the outer function.