Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5. (g∘ƒ)(0)

Verified step by step guidance

1. Step 1: Understand the notation \((g \circ f)(x)\), which means the composition of functions \(g\) and \(f\). This is equivalent to \(g(f(x))\).
2. Step 2: Start by finding \(f(0)\) using the function \(f(x) = 2x - 3\). Substitute \(x = 0\) into \(f(x)\).
3. Step 3: Calculate \(f(0) = 2(0) - 3\). Simplify this expression to find the value of \(f(0)\).
4. Step 4: Use the result from Step 3 as the input for the function \(g(x) = -x + 3\). Substitute \(f(0)\) into \(g(x)\) to find \(g(f(0))\).
5. Step 5: Calculate \(g(f(0)) = -f(0) + 3\). Simplify this expression to find the value of \((g \circ f)(0)\).

Key Concepts

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. This process is essential for evaluating composite functions, as it requires substituting the output of one function into another.

Evaluating Functions

Evaluating a function means finding the output value for a given input. For example, to evaluate ƒ(0) for the function ƒ(x)=2x-3, you substitute 0 for x, resulting in ƒ(0)=2(0)-3=-3. This step is crucial in function composition, as you need to compute the value of the inner function before applying the outer function.

Linear Functions

Linear functions are mathematical expressions of the form f(x) = mx + b, where m is the slope and b is the y-intercept. Both ƒ(x)=2x-3 and g(x)=-x+3 are linear functions, which means their graphs are straight lines. Understanding their properties, such as slope and intercepts, helps in visualizing and solving problems involving these functions.