Problem 24 - Beginning and Intermediate Algebra

Question

Let \(f(x)=x^{2}+7 x-9\) and \(g(x)=x+2 .\) Find a) \((g \circ f)(x)\) b) \(\quad(f \circ g)(x)\) c) \((g \circ f)(3)\)

Step-by-Step Solution

Answer

a) \( (g \circ f)(x) = x^2 + 7x - 7 \) b) \( (f \circ g)(x) = x^2 + 11x + 9 \) c) \( (g \circ f)(3) = 23 \)

Step 1: Understand function composition

Function composition, denoted (g ◦ f)(x) or g(f(x)), means that you apply the function g to the result of applying the function f to x. In other words, you first apply f(x) and then use the result as input for g(x). The same concept applies when we compute (f ◦ g)(x) or f(g(x)).

Step 2: Compute (g ◦ f)(x)

To find (g ◦ f)(x), we substitute f(x) in the input of g(x). For the given functions, we apply g(x) to f(x): (g ◦ f)(x) = g(f(x)) = g(x^2 + 7x - 9) And now substituting: (g ◦ f)(x) = (x^2 + 7x - 9) + 2 = x^2 + 7x - 7 Thus, (g ◦ f)(x) = x^2 + 7x - 7.

Step 3: Compute (f ◦ g)(x)

Similarly, for computing (f ◦ g)(x), we substitute g(x) in the input of f(x). For the given functions, we apply f(x) to g(x): (f ◦ g)(x) = f(g(x)) = f(x + 2) And now substituting: (f ◦ g)(x) = (x+2)^2 + 7(x+2) -9 Expanding and simplifying: (f ◦ g)(x) = x^2 + 4x + 4 + 7x + 14 - 9 = x^2 + 11x + 9 Thus, (f ◦ g)(x) = x^2 + 11x + 9.

Step 4: Compute (g ◦ f)(3)

To find (g ◦ f)(3), we substitute x = 3 in the expression we derived for (g ◦ f)(x) earlier: (g ◦ f)(3) = (3)^2 + 7(3) - 7 = 9 + 21 - 7 = 23 Thus, (g ◦ f)(3) = 23. In summary: a) (g ◦ f)(x) = x^2 + 7x - 7 b) (f ◦ g)(x) = x^2 + 11x + 9 c) (g ◦ f)(3) = 23

Key Concepts

Algebra

Algebra is a key part of mathematics that involves the use of symbols and letters to represent numbers and quantities in formulas and equations. It's like a universal language that allows us to describe patterns and solve problems. When dealing with functions, algebra allows us to find the relationship between different sets of numbers.

One of the fundamental concepts in algebra is simplifying expressions. This process involves combining like terms and using operation rules to make equations easier to work with or to act upon functions. Understanding these basics gives us the power to handle more complicated tasks, such as evaluating functions, like what we see in function composition.

• It enables us to manipulate variables.
• Provides a systematic way of solving equations.
• It helps in understanding the behavior of functions.

As we dive into function composition in algebra, we use multiple functions in sequence to achieve a specific outcome, illustrating the flexibility and power of algebraic operations.

Polynomials

Polynomials are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. They form the building blocks of algebra and are fundamental to understanding how functions behave.

In the given exercise, we have a polynomial function, such as \(f(x) = x^2 + 7x - 9\). Here's what makes polynomials essential:

• Symmetry: Polynomials can be reflective around certain points, making them easier to analyze graphically.
• Smooth Graphs: The graphs of polynomial functions are continuous and smooth, offering insight into their behavior.
• Predictable Behavior: The degree of the polynomial (the highest power of the variable) tells us a lot about the function's end behavior, roots, and number of turns.

By understanding polynomials, we can predict and compute the results of polynomial functions when combined, such as in composite functions like \((f \circ g)(x)\). This involves substituting one polynomial into another, demonstrating how polynomials form networks of relationships in algebra.

Composite Functions

Composite functions involve applying one function to the result of another function, often denoted as \((g \circ f)(x)\), meaning \(g(f(x))\). This concept is crucial in mathematics because it introduces the idea of layering processes upon processes, leading to complex behavior from simple functions.

In our example, after solving each component separately, we find:

• \((g \circ f)(x) = x^2 + 7x - 7\) is the result of applying \(g(x)\) after \(f(x)\) has done its work.
• \((f \circ g)(x) = x^2 + 11x + 9\) shows the effect of plugging \(g(x)\) into \(f(x)\).

Composite functions enable us to explore relationships between variables in a manageable way by encapsulating multiple processes within a single formula. This technique allows simplification and analysis of the layers of mathematical problems, offering deeper insights and solutions.

Understanding the mechanics of composition not only helps solve exercises but strengthens overall mathematical intuition by showing how different operations can build upon each other to create new expressions or solve complex problems.