Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5.
$open paren f with hook composed with g close paren times 4$

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Understand that the notation $(ƒ \circ g)(4)$ means you first apply the function $g$ to 4, then apply the function $ƒ$ to the result of $g(4)$.

Calculate $g(4)$ by substituting $x = 4$ into the function $g(x) = -x + 3$. This gives $g(4) = -4 + 3$.

Simplify the expression for $g(4)$ to find its value.

Next, take the result from $g(4)$ and substitute it into the function $ƒ(x) = 2x - 3$. This means you calculate $ƒ(g(4)) = 2 \times g(4) - 3$.

Simplify the expression for $ƒ(g(4))$ to find the final value of $(ƒ \circ g)(4)$.

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주요 개념

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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. This concept is essential for understanding how to combine functions and find composite values.

Evaluating Functions

Evaluating a function means substituting a specific input value into the function's formula and simplifying to find the output. For example, to find g(4), replace x with 4 in g(x) = -x + 3 and simplify. Accurate evaluation is crucial for solving composite functions.

Linear Functions

Linear functions have the form f(x) = mx + b, where m and b are constants. They produce straight-line graphs and are easy to evaluate and compose. Understanding their structure helps in quickly calculating values and interpreting function behavior.

Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5. (ƒ∘g)(4)(ƒ∘g)(4)