Problem 41 - Elementary and Intermediate Algebra

Question

Let \(f(x)=2 x+1\) and \(g(x)=x^{2}-1 .\) Find each of the following. $$ (f \circ g)\left(\frac{1}{2}\right) $$

Step-by-Step Solution

Answer

\((f \circ g)(\frac{1}{2}) = -\frac{1}{2}.\)

Step 1: Understand the notation

The notation \((f \circ g)(x)\) represents the composition of functions, which means applying function \(g\) first and then applying function \(f\) to the result of \(g\). Thus, \((f \circ g)(x) = f(g(x))\).

Step 2: Compute \(g\left(\frac{1}{2}\right)\)

Substitute \(x = \frac{1}{2}\) into \(g(x) = x^2 - 1\):\[g\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^2 - 1 = \frac{1}{4} - 1 = -\frac{3}{4}.\]

Step 3: Compute \(f(g(\frac{1}{2}))\)

Use the result from Step 2, \(g\left(\frac{1}{2}\right) = -\frac{3}{4}\), and substitute it into \(f(x) = 2x + 1\):\[f\left(-\frac{3}{4}\right) = 2\left(-\frac{3}{4}\right) + 1 = -\frac{6}{4} + 1 = -\frac{3}{2} + 1 = -\frac{1}{2}.\]

Key Concepts

Function Notation

Function notation is a mathematical shorthand used to express the relationship between a set of inputs and their corresponding outputs. In simpler terms, instead of writing 'y' equals something, you use function notation like 'f(x)' to represent the function. Here, 'f(x)' means the output of the function 'f' when the input is 'x'. Function notation allows for a clear and concise way to depict these relationships. For example:

• The function \(f(x) = 2x + 1\) shows that to get the output, you multiply the input by 2 and then add 1.
• Similarly, the function \(g(x) = x^2 - 1\) tells us that you first square the input and then subtract 1.

Understanding this notation is key to solving problems involving functions, especially when dealing with more complex compositions.

Quadratic Functions

Quadratic functions are a type of polynomial function represented by functions like \(g(x) = x^2 - 1\). The highest power of the variable in a quadratic function is 2, giving it the name 'quadratic'. This function creates a characteristic parabola shape when graphed. Key aspects of quadratic functions include:

• The leading coefficient, which influences the direction in which the parabola opens (upward if positive, downward if negative).
• The constant term, which helps determine the vertex's position along the y-axis.

Quadratic equations are essential because they often model real-world phenomena, such as projectile motion. Understanding how to evaluate these functions is crucial for solving more complex problems like function compositions.

Linear Functions

Linear functions, such as \(f(x) = 2x + 1\), are functions where the variable has an exponent of 1, meaning they're of the first degree. These functions create straight lines when graphed, which is why they are called linear. Linear functions exhibit two main properties:

• Slope, which indicates how steep the line is and in which direction it moves.
• Y-intercept, which is the point where the line crosses the y-axis.

This simplicity makes linear functions easy to work with and interpret. When combined with other types of functions in compositions, such as quadratic functions, they can add layers of complexity and lead to interesting results. Mastery of linear functions is fundamental in various aspects of mathematics and their applications in real-life problems.