Problem 28 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find (a) \(f \circ g\) and (b) \(g \circ f\). $$f(x)=5 x+8, g(x)=2 x^{2}-1$$

Step-by-Step Solution

Answer

\((a) f \circ g = 10x^{2} +3\), and \((b) g \circ f = 50x^{2} + 80x + 127\).

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

We first need to find \(f \circ g\). The composition \(f \circ g\) is defined as \(f(g(x))\). So, we replace every 'x' in \(f(x)=5x+8\) with \(g(x)=2x^{2}-1\). This gives us \(f(g(x)) = 5(2x^{2}-1) + 8\).

Step 2: Simplify \(f \circ g\)

Now, we simplify the expression obtained in step 1. Distributing 5 into the parenthesis, we get \(f(g(x)) = 10x^{2} - 5 +8\), simplifying the constants yields, \(f(g(x)) = 10x^{2} +3\).

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

Secondly, we need to find \(g \circ f\). The composition \(g \circ f\) is defined as \(g(f(x))\). So, we replace every 'x' in \(g(x)=2x^{2}-1\) with \(f(x)=5x+8\). This gives us \(g(f(x)) = 2(5x + 8)^{2} -1\).

Step 4: Simplify \(g \circ f\)

Finally, we simplify the expression obtained in step 3. Expanding the squared term, and simplifying, we get \(g(f(x)) = 2(25x^{2} + 40x + 64) -1\). On further simplification, this yields: \(g(f(x)) = 50x^{2} + 80x + 127\).

Key Concepts

Composite Functions

Composite functions involve the combination of two functions where the output of one function becomes the input of the next. This process is called function composition. There are different notations used to denote composite functions. Consider two functions, say, \(f(x)\) and \(g(x)\). The composition \(f \circ g\) is read as \(f\) composed with \(g\), and is defined as \(f(g(x))\). Similarly, \(g \circ f\) is written as \(g(f(x))\). Essentially, function composition allows us to create more complex functions from simpler ones. This is useful in various applications in mathematics. For example, if \(f(x)\) represents a process and \(g(x)\) another, the composition \(f \circ g\) might model the result of applying both processes successively.

Function Notation

Function notation is a way to express a relationship between inputs and outputs in a precise and systematic manner. In mathematics, a function \(f\) assigns an output \(f(x)\) to each input \(x\). This is indicated using a function rule, which is typically in the form \(f(x) = expression\). For example, for \(f(x) = 5x + 8\), the function rule expresses a linear relationship between \(x\) and \(f(x)\). When dealing with composite functions, function notation is invaluable as it allows us to specify which function is applied first. For example, with \(f(g(x))\), \(g\) is applied to \(x\) first, and the result is then used as an input for \(f\). Understanding notation helps avoid errors and ensures accuracy in function manipulation.

Quadratic Function

Quadratic functions are a type of polynomial function where the highest degree of the variable \(x\) is 2. The standard form of a quadratic function is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). For example, in the exercise, \(g(x) = 2x^2 - 1\) is a quadratic function. These functions are characterized by their parabolic graphs, which may open upward or downward depending on the sign of \(a\). Quadratic functions are prevalent in real-life scenarios, such as modeling projectile motion. In function composition, performing operations with quadratic functions requires careful algebraic manipulation, like expanding and simplifying expressions, as seen in solving \(g(f(x))\). Understanding quadratics aids in identifying these characteristics and effectively interpreting graphs.

Linear Function

Linear functions describe a constant rate of change, represented by straight lines on a graph. They follow the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In the given exercise, \(f(x) = 5x + 8\) is a linear function. The slope \(m = 5\) indicates that for every unit increase in \(x\), \(f(x)\) increases by 5 units, and the intersect \(b = 8\) shows the point where the line crosses the y-axis. Linear functions are straightforward and serve as building blocks in function composition. When composing with other function types, like quadratic functions, linear functions affect the output in a predictable yet impactful way. For instance, in \(f(g(x))\), the linear function alters the quadratic output into a more complex form, emphasizing the versatility of linear functions in composite expressions.