Problem 73 - College Algebra

Question

If equations for functions \(f\) and \(g\) are given, describe two ways to find \((f-g)(3)\).

Step-by-Step Solution

Answer

Two methods to find \((f-g)(3)\): 1. Evaluate \(f(3)\) and \(g(3)\) separately and then subtract. 2. Form the function \(h(x) = f(x) - g(x)\) and then substitute \(x=3\).

Step 1: Method 1: Evaluate then Subtract

In this method, first calculate \(f(3)\) and \(g(3)\) separately. After finding these two values, subtract the value of \(g(3)\) from \(f(3)\). This gives the result of \((f-g)(3)\).

Step 2: Method 2: Subtract then Evaluate

In the second method, first form the function \(h(x) = f(x) - g(x)\). Then substitute \(x = 3\) into this new function \(h(x)\). The result will be the same as in the first method, that is, the value of \((f-g)(3)\).

Key Concepts

Understanding Function Evaluation

Function evaluation is the process of finding the output of a function for a specific input. To do this, simply substitute the input value into the function's equation and perform the arithmetic operations involved.

• Start by identifying the function and the input value. For example, if you have a function \(f(x) = 2x + 1\) and you need to evaluate it at \(x = 3\), replace \(x\) with 3: \(f(3) = 2(3) + 1 = 7\).
• Perform all calculations according to the order of operations: parentheses, exponents, multiplication and division (from left to right), and then addition and subtraction (from left to right).
• For multiple functions, perform the evaluation separately for each, using the given input value.

This simple substitution method is used in both function subtraction and composite functions, making it a foundational skill for more complex operations.

Mastering Function Subtraction

Function subtraction involves creating a new function by taking each output of one function and subtracting it from the output of another function at the same input.

• Begin by identifying the two functions you want to subtract, for example, \(f(x) = 2x\) and \(g(x) = x + 3\).
• Create a new function \(h(x)\) defined as \(h(x) = f(x) - g(x)\). In our case, this would be \(h(x) = 2x - (x + 3)\), which simplifies to \(h(x) = x - 3\).
• In the context of subtraction then evaluation, you can substitute any specific input into this new function to find the result, such as \(h(3) = 3 - 3 = 0\).

This operation is useful for directly comparing two functions and understanding differences in their behavior at specific points.

Exploring Composite Functions

Composite functions involve combining two functions such that the output of one function becomes the input of another. This can be represented as \((f \circ g)(x)\), which means \(f(g(x))\).

• Begin by evaluating the inner function \(g(x)\) for a specific value. For example, if \(g(x) = x+2\) and you want to find \((f \circ g)(3)\), calculate \(g(3) = 3 + 2 = 5\).
• Use the result from the inner function, \(g(3) = 5\), as the input for the outer function \(f(x)\). If \(f(x) = 2x\), then \(f(g(3)) = f(5) = 2(5) = 10\).
• Through this process, you effectively evaluate the composite function by chaining the operations of two functions based on their definitions.

This method highlights the sequence and nesting of functions, a crucial concept for analyzing complex function relationships.