Problem 52 - Precalculus : Building Concepts and Connections

Question

In Exercises \(49-66,\) let \(f(x)=x^{2}+x, g(x)=\sqrt{x},\) and \(h(x)=-3 x\) Evaluate each of the following. $$(f \circ h)(-1)$$

Step-by-Step Solution

Answer

The result of the function composition \((f \circ h)(-1)\) is \(12\).

Step 1: Define the Functions

First, note the given functions: \(f(x)=x^{2}+x,\) \(h(x)=-3 x\). Your task is to find the composition \((f \circ h)(-1)\), which means to plug \(h(-1)\) into \(f(x)\).

Step 2: Find \(h(-1)\)

Substitute \(x = -1\) into the function \(h(x)\) to get \(h(-1) = -3(-1) = 3\).

Step 3: Substitute \(h(-1)\) in \(f(x)\)

Now substitute \(h(-1) = 3\) into \(f(x)\) to find \(f(h(-1)) = f(3)\). Using \(f(x) = x^2+x\), then \(f(3) = 3^{2}+3 = 9+3 = 12\).

Step 4: Simplification

So, the composition \((f \circ h)(-1)\) simplifies to \(12\).

Key Concepts

Quadratic Functions

Quadratic functions are essential in algebra and are characterized by their standard form: \(f(x) = ax^2 + bx + c\). A key feature of quadratic functions is their parabolic shape when graphed, which means the graph forms a U-shape or an inverted U-shape, depending on the sign of \(a\).

For example, in the function \(f(x) = x^2 + x\), we set \(a = 1\), \(b = 1\), and \(c = 0\). This particular form implies that the parabola opens upwards, as \(a\) is positive. Quadratic functions are known for having a vertex, an axis of symmetry, and they exhibit either a minimum or a maximum point depending on the direction of the parabola.

Quadratic functions are used to model various real-life scenarios, such as projectile motion or the area of objects. Understanding how to manipulate these functions, like finding the vertex or determining the roots, is an integral part of algebra and precalculus learning.

Linear Functions

Linear functions form the foundation of algebra and are typically represented by the equation \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept.

In the context of our example, we are given a linear function \(h(x) = -3x\). This function is straightforward as it demonstrates a direct relationship where the slope \(m = -3\) indicates that the line decreases by 3 units in the y-direction for every 1 unit increase in the x-direction.

This kind of function results in a straight line graph, and it is widely applicable, ranging from predicting trends to solving simple equations. Linear functions are distinguishable by their constant rate of change, making them a pivotal concept in understanding more complex functions.

Radical Functions

Radical functions include a variable within a radical sign, usually involving square roots. The simplified form can be \(g(x) = \sqrt{x}\), as seen in our example.

This kind of function is uncommon because the domain is restricted to non-negative values of \(x\), ensuring that the real number outcomes of the square root are valid. The graph of a simple radical function like \(g(x) = \sqrt{x}\) is half of a sideways parabola, gradually increasing without bound as \(x\) approaches infinity.

Understanding the behavior of radical functions is crucial, as they frequently appear in solutions to quadratic equations and in various scientific formulas, particularly in physics and engineering contexts. Mastering these functions can support a deeper comprehension of mathematical principles and real-world applications.