Problem 31 - Algebra 2

Question

Let \(f(x)=x^{2}\) and \(g(x)=x-3 .\) Find each value or expression. $$ (g \circ f)(-2) $$

Step-by-Step Solution

Answer

The value of \((g \circ f)(-2)\) is 1.

Step 1: Apply Function \(f\)

The function \(f(x) = x^2\), so the first task is to compute \(f(-2)\). Substituting -2 into this function gives \((-2)^2 = 4\).

Step 2: Apply Function \(g\)

The result from the first step is now the input to the function \(g\). The function \(g(x) = x - 3\), so substituting the output from step 1 into this function gives \(4 - 3 = 1\).

Step 3: Final Result

The final result of \((g \circ f)(-2)\) is the output from step 2, which is 1.

Key Concepts

Understanding Quadratic Functions

A quadratic function, often expressed in the form of \( f(x) = ax^2 + bx + c \), represents a polynomial function of degree 2. It often forms a parabola when graphed, which opens upward if the leading coefficient \(a\) is positive and downward if \(a\) is negative.

In this particular exercise, the quadratic function is simplified to \( f(x) = x^2 \). This means every input value \(x\) is squared to determine the output. Quadratic functions can be used to model various real-world phenomena such as projectile motion and area calculations.

• The function has a vertex, which in this simple form \( f(x) = x^2 \) is at the point (0, 0).
• It is symmetric around the y-axis, which means that \( f(-x) = f(x) \) for every \(x\).
• The graph expands upwards which makes this an example of a convex function.

By understanding how quadratic functions manipulate their input values, you can easily compute outputs for given inputs, such as \( f(-2) = (-2)^2 = 4 \). This is a crucial step in solving composed function problems.

Breaking Down Linear Functions

Linear functions are one of the simplest types of functions in mathematics. The general form of a linear function is \( g(x) = mx + b \), where \(m\) represents the slope and \(b\) the y-intercept.

In this textbook exercise, we have a linear function \( g(x) = x - 3 \). Here, the coefficient of \(x\) is 1, indicating a slope of 1, meaning the function increases by 1 unit along the y-axis for each 1 unit increase on the x-axis.

• It has a constant slope, plotting as a straight line on a graph.
• In terms of transformation, this function translates every output of its input \(x\) down by 3 units because of the \(-3\).
• Linear functions are important for modeling relationships where things change at a constant rate.

Understanding linear functions allows us to easily evaluate them by substituting the input value. For instance, substituting the output of the quadratic function, we find \( g(f(x)) = 4 - 3 = 1 \) with this linear transformation.

Mastering Function Evaluation

Function evaluation is an essential mathematical process. It involves substituting a specific value into a function and computing the result. Understanding this process allows you to execute operations correctly in function composition.

To evaluate a function, you replace the variable with a given number or expression. In the original exercise, we first evaluated \(f(x) = x^2\) at \(x = -2\) to get \(f(-2) = 4\). This output then became the input for the subsequent evaluation of \(g(x) = x - 3\), hence \(g(4) = 1\).

• Start by identifying the function and the input value.
• Substitute the input into the function.
• Simplify the expression to find the output.

Function evaluation becomes particularly crucial during function composition, where the output of one function becomes the input of another. Mastery of function evaluation ensures you handle each step accurately, leading to the correct result.