Problem 22 - College Algebra

Question

Find: a. \((f \circ g)(x)\) b. \(\left(g^{\circ} f\right)(x)\) c. \((f \circ g)(2)\) $$f(x)=7 x+1, g(x)=2 x^{2}-9$$

Step-by-Step Solution

Answer

\((f \circ g)(x) = 14x^2 - 62; (g^{\circ} f)(x) = 98x^2 + 28x -7 ; (f \circ g)(2) = -6 \)

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

First, substitute \(g(x)\) into \(f(x)\). This gives \(f(g(x)) = 7(2x^2 - 9) + 1 = 14x^2 - 63 + 1 = 14x^2 -62 \)

Step 2: Compute \(g^{\circ} f(x)\)

Now, substitute \(f(x)\) into \(g(x)\). This will give \(g(f(x)) = 2(7x + 1)^2 - 9 = 2(49x^2 + 14x + 1) - 9 = 98x^2 + 28x +2 -9 = 98x^2 + 28x -7 \)

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

Finally, substitute \(x = 2\) into \(f \circ g(x)\). This will yield \((f \circ g)(2) = 14(2^2) - 62 = 14*4 - 62 = 56 - 62 = -6 \)

Key Concepts

Function Operations

Function operations are processes that combine two or more functions in ways that produce a new function. In algebra, the most common operations are addition, subtraction, multiplication, division, and composition. The operation that is most intriguing among these is the composition of functions, denoted by \(f \circ g\). This operation involves applying one function to the results of another function.

For example, if you have two functions, \(f(x)\) and \(g(x)\), the composition \(f \circ g(x)\) means you first apply \(g(x)\) and then apply \(f(x)\) to the result. It's like a 'function of a function'. The order in which the functions are composed matters, as \(f \circ g(x)\) generally yields a different result than \(g \circ f(x)\). In our case, we've seen that the composition requires us to substitute \(g(x)\) into \(f(x)\), and then simplify to find the new composite function.

Algebraic Functions

Algebraic functions are functions that can be expressed using arithmetic operations and can involve powers and roots of the variable \(x\). They are the building blocks of algebra and form the basis for advanced topics in mathematics and sciences. Algebraic functions are represented by equations involving polynomials, rational expressions, and radicals, among others.

In this instance, \(f(x) = 7x + 1\) and \(g(x) = 2x^2 - 9\) are both algebraic functions. The first function, \(f\), is a linear function, as it graphs a straight line with a slope of 7 and a y-intercept of 1. The second function, \(g\), is a quadratic function, which graphs a parabola. It has a leading coefficient of 2, which determines the degree and the 'width' of the parabola, and a constant term of -9, which affects its vertical position on the graph.

Function Evaluation

Function evaluation involves finding the value of a function for a particular value of \(x\). When you evaluate a function, you simply substitute the number for \(x\) and calculate the result. It's a fundamental skill in algebra since it allows you to understand the behavior of functions at specific points.

Whenever you're given a function like \(f(x)\) or \(g(x)\), and you need to evaluate it at a point such as \(x = 2\), you replace every \(x\) in the function's formula with 2 and then simplify. For example, by evaluating \(f \circ g(2)\), we calculate the value of \(g(x)\) at \(x = 2\) and then plug this value into the function \(f(x)\). In the exercise, this evaluation showed us that the value of the composed function \(f \circ g\) at \(x = 2\) is -6.