Question

If \(f(x)=2 x+3\) and \(g(x)=9 x+6\), then find \(g[f(x)]-f[g(x)]\) (a) 18 (b) 22 (c) 20 (d) none of these

Short answer

Answer: (a) 18

Step-by-step solution

Find g[f(x)]

To find the composite function \(g[f(x)]\), we need to substitute \(f(x)\) into \(g(x)\). So, replace every \(x\) in the expression for \(g(x)\) with the expression for \(f(x)\): \(g[f(x)] = g(2x + 3) = 9(2x + 3) + 6\) Now, perform the operations to simplify the expression: \(g[f(x)] = 18x + 27 + 6 = 18x + 33\)

Find f[g(x)]

To find the composite function \(f[g(x)]\), we need to substitute \(g(x)\) into \(f(x)\). So, replace every \(x\) in the expression for \(f(x)\) with the expression for \(g(x)\): \(f[g(x)] = f(9x + 6) = 2(9x + 6) + 3\) Now, perform the operations to simplify the expression: \(f[g(x)] = 18x + 12 + 3 = 18x + 15\)

Calculate g[f(x)] - f[g(x)]

To find \(g[f(x)] - f[g(x)]\), we will subtract the expressions obtained in steps 1 and 2: \(g[f(x)]-f[g(x)] = (18x + 33) - (18x + 15)\) Now, subtract the terms: \(g[f(x)]-f[g(x)] = 18x + 33 - 18x - 15 = 18 - 18x + 33 - 15\) Combine like terms: \(g[f(x)]-f[g(x)] = 18\)

Compare the result with the options

Our result is 18, which matches option (a). Therefore, the correct answer is: \((a) 18\)

Key concepts

Function Operations

When dealing with functions, it is essential to understand various operations we can perform on them. These function operations include addition, subtraction, multiplication, and division of functions. Another crucial operation is composition, where one function is plugged into another.

Let's break down what an operation looks like with the given functions:

• Addition: If we have functions \( f(x) \) and \( g(x) \), the operation would be \((f+g)(x) = f(x) + g(x)\).
• Subtraction: Subtracting functions is represented as \((f-g)(x) = f(x) - g(x)\).
• Multiplication and division work similarly, but great care must be taken, especially with division, to avoid division by zero.

All these operations aim to create a new function or expression by altering the given ones. When working with composite functions, we deal with a more complex operation that involves inserting one function into another.

Algebra

Algebra is the backbone of solving problems involving function operations. It involves using various algebraic rules and techniques to simplify expressions and solve equations.

For instance, when solving \(g[f(x)] = g(2x + 3)\), you replace every instance of \(x\) in \(g(x)\) with \(f(x)\). This substitution allows you to manipulate the functions algebraically, transforming them into a simpler form: \(g[f(x)] = 18x + 33\).

Algebra also helps in simplifying the results when performing operations with composite functions. For example, when you have \(g[f(x)] - f[g(x)]\), you subtract the expressions: \((18x+33) - (18x+15)\). Applying algebraic rules like combining like terms simplifies this to \(18\), showing the power of algebra in problem-solving.

Function Composition

Function composition is a slightly more advanced operation in the world of functions. It involves creating a new function by putting one function inside another, denoted as \( (g \circ f)(x) = g(f(x)) \).

Composition is not merely addition or subtraction where functions combine directly. Instead, you take the output of one function \( f(x) \) and use it as the input for another \( g(x) \). In our example, \(g[f(x)]\) is an illustration of this concept, where you begin by replacing each \(x\) in the expression for \(g(x)\) with the expression you find for \(f(x)\). .

The process demands careful substitution and simplification, reflecting on why understanding each function individually first is critical. With correct functional composition, you can interpret and solve more complex function-based problems effectively.