Problem 21 - College Algebra

Question

Find: a. \((f \circ g)(x)\) b. \(\left(g^{\circ} f\right)(x)\) c. \((f \circ g)(2)\) $$f(x)=4 x-3, g(x)=5 x^{2}-2$$

Step-by-Step Solution

Answer

\((f \circ g)(x) = 20x^2-11\), \((g \circ f)(x) = 80x^2-120x+43\), \((f \circ g)(2) = 69\).

Step 1: Find \((f \circ g)(x)\)

To find \((f \circ g)(x)\), replace every instance of \(x\) in function \(f\) with the function \(g(x)\), to get \(f(g(x))\). So \(f(g(x)) = 4(5x^2 - 2) - 3\). Distribute and simplify to get \(20x^2 -8 -3\), which simplifies further to \(20x^2-11\). So \((f \circ g)(x) = 20x^2-11\).

Step 2: Find \((g \circ f)(x)\)

In a similar way, find \((g \circ f)(x)\) by replacing every instance of \(x\) in function \(g\) with the function \(f(x)\), to get \(g(f(x))\). So \(g(f(x)) = 5(4x - 3)^2 - 2\). This simplifies to \(5(16x^2-24x+9)-2\), and further to \(80x^2-120x+45-2\), which simplifies to \(80x^2-120x+43\). So \((g \circ f)(x) = 80x^2-120x+43\).

Step 3: Find \((f \circ g)(2)\)

To find \((f \circ g)(2)\), replace \(x\) with \(2\) in the function \(f(g(x))\) derived in step 1. So let's compute \(f(g(2)) = 20(2)^2-11 = 80-11 = 69\). So \((f \circ g)(2) = 69\).

Key Concepts

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number exponents and combined using addition, subtraction, and multiplication. These functions can be represented in the standard form, with the highest degree term written first. For example, a polynomial function like

• \(f(x) = ax^n + bx^{n-1} + ... + k\)
• where \(a, b,\) and \(k\) are constants and \(n\) is a non-negative integer.

Understanding polynomial functions is crucial because they appear frequently in algebra. They can describe a wide range of real-world phenomena, from simple to complex relationships. In the exercise, we looked into two polynomial functions:

• Function \(f(x) = 4x - 3\) is a linear polynomial of degree 1.
• Function \(g(x) = 5x^2 - 2\) is a quadratic polynomial of degree 2.

Linear polynomials form straight lines when graphed, whereas quadratic polynomials create parabolas. Identifying the type of polynomial helps in predicting its behavior and graph.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent mathematical relationships. They can be as simple as a single term or as complex as multiple terms with a mix of operations.An understanding of algebraic expressions allows for manipulating and simplifying these to solve various mathematical problems. The key steps involve combining like terms and simplifying coefficients.For example, in the exercise solution, when simplifying \(f(g(x)) = 4(5x^2 - 2) - 3\),

1. we first distribute the \(4\) across the polynomial \(5x^2 - 2\), resulting in the terms \(20x^2 - 8\).
2. Then, the constant term \(-3\) is added, leading to the final simplified form \(20x^2 - 11\).

By understanding how to work with algebraic expressions, you can more easily solve or simplify equations, helping with everything from basic problems to advanced calculus.

Substitution in Functions

Substitution in functions is a method used to evaluate composite functions or find specific values within functions. It involves replacing the variable in a function with another expression or a particular number.In the given exercise, to find \((f \, \circ \, g)(x)\),

• We substitute \(g(x) = 5x^2 - 2\) into every \(x\) of the function \(f(x) = 4x - 3\). This creates the new composition \(f(g(x)) = 4(5x^2 - 2) - 3\).
• For the specific value calculation \((f \, \circ \, g)(2)\), we replace \(x\) with \(2\) in \(f(g(x))\). So,
  • First calculate \(g(2) = 5(2)^2 - 2 = 18\),
  • then substitute \(18\) into \(f(x)\), resulting in \(f(18) = 4 \times 18 - 3 = 69\).

By understanding substitution, you gain the ability to simplify complex expressions and solve equations, enhancing your problem-solving skills significantly.