Problem 68 - Algebra 2

Question

For each pair of functions, find \(f(g(x))\) and \(g(f(x))\) $$ f(x)=\frac{x+5}{2}, g(x)=x^{2} $$

Step-by-Step Solution

Answer

The composed functions are f(g(x)) = \(\frac{x^{2}+5}{2}\) and g(f(x)) = \(\left(\frac{x+5}{2}\right)^{2}\).

Step 1: Calculate f(g(x))

First, replace every x in f(x) with g(x). The function f(x) is \(\frac{x+5}{2}\), so f(g(x)) becomes \(\frac{g(x)+5}{2}\). Plug the formula for g(x), which is \(x^{2}\), into the equation, yielding \(\frac{x^{2}+5}{2}.\)

Step 2: Calculate g(f(x))

Now, replace every x in g(x) with f(x). The function g(x) is \(x^{2}\), so g(f(x)) becomes \(f(x)^{2}.\). Now replace f(x) with its formula, i.e., \(\frac{x+5}{2}\). Squaring this yields \(\left(\frac{x+5}{2}\right)^{2}.\).

Key Concepts

Understanding Function Composition

Function composition is a crucial concept in mathematics. It involves creating a new function by applying one function to the results of another. When you see a notation like \(f(g(x))\) or \(g(f(x))\), it means you're evaluating one function within another. This process allows us to explore more complex relationships between variables beyond simple functions.
To perform function composition, follow these steps:

• Identify the inner function and the outer function. In \(f(g(x))\), \(g(x)\) is the inner function and \(f\) is the outer function.
• Substitute the entire inner function into every instance of the variable in the outer function.
• Simplify the expression as much as possible.

Mastering function composition can aid in understanding transformations and complex operations in mathematics. It is widely used in calculus and other advanced topics.

What is a Quadratic Function?

A quadratic function is a polynomial function of degree 2. Its general form is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(aeq 0\). The graph of a quadratic function is a parabola, which opens either upwards or downwards.
Key features of quadratic functions:

• The vertex of the parabola is the maximum or minimum point, depending on the direction it opens.
• The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.
• The roots of the quadratic equation, if real, are where the parabola intersects the x-axis.

In the given exercise, \(g(x) = x^2\) is a simple quadratic function with its graph opening upwards, having its vertex at the origin \((0, 0)\). Understanding quadratic functions helps in analyzing a variety of mathematical models in different fields like physics and engineering.

An Introduction to Linear Functions

Linear functions are the simplest type of function. They are represented by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This equation yields a straight line when graphed.
Important aspects include:

• The slope \(m\) determines the steepness and direction of the line; positive slopes rise, while negative slopes fall.
• The y-intercept \(b\) indicates where the line crosses the y-axis.
• Linear functions have a constant rate of change, making them crucial for understanding proportionality and prediction in real-world situations.

In our exercise, the function \(f(x) = \frac{x+5}{2}\) can be rearranged into a linear form \(y = \frac{1}{2}x + \frac{5}{2}\). This description illustrates how linear functions provide simple yet powerful tools for mapping relationships between variables.