Problem 53 - 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. $$(h \circ g)(4)$$

Step-by-Step Solution

Answer

\((h \circ g)(4) = -6\)

Step 1: Find g(4)

First, we need to find the value of the function \(g\) at \(x=4\). Since \(g(x)=\sqrt{x}\), we substitute \(x=4\) into the function: \(g(4) = \sqrt{4} = 2.

Step 2: Substitute g(4) into h(x)

Next, we substitute the result of \(g(4)\) into the function \(h\). Since \(h(x)=-3x\), substituting the values in we obtain:\(h(g(4)) = h(2) = -3*2 = -6\).

Step 3: Write down the final answer

Finally, we write down the final answer. The value of \((h \circ g)(4)\) is -6.

Key Concepts

Quadratic Functions

Quadratic functions are fundamental in mathematics, especially in algebra and calculus. They are polynomial functions of degree two, typically represented in the form:\[ f(x) = ax^2 + bx + c \]Here, \(a\), \(b\), and \(c\) are constants, with the restriction that \(a eq 0\). The graph of a quadratic function is a parabola.

• **Vertex**: The highest or lowest point on the graph, depending on the parabola's orientation, is called the vertex.
• **Axis of Symmetry**: This is a vertical line that passes through the vertex and divides the parabola into mirror images.
• **Roots**: The points where the parabola intersects the x-axis. They can be found using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Quadratics are everywhere, from physics to finance, modeling anything from projectile motion to profit maximization. Understanding their behavior helps in solving complex mathematical problems with ease.

Square Root Function

The square root function is expressed as \( g(x) = \sqrt{x} \). It is a unique and continuous function that yields a non-negative number, representing the principal square root of \( x \).

• **Domain**: The set of all real numbers greater than or equal to zero. This is because square roots of negative numbers are not defined in the realm of real numbers.
• **Range**: All non-negative numbers. A square root function begins at the origin (if written as \( \sqrt{x} \)) and extends indefinitely to the right along the x-axis.
• **Graph**: It is a standard curve that starts at the origin and curves upwards to the right, gradually becoming less steep.

The square root function is vital in many areas such as solving quadratic equations and manipulating formulas. Its simplicity, yet mathematical importance, makes it an essential topic in mathematics.

Linear Functions

Linear functions are one of the simplest yet most important classes of functions in mathematics. Any function in the form \[ h(x) = mx + b \] is called a linear function, where \( m \) is the slope of the line, and \( b \) is the y-intercept.

• **Slope**: It indicates the rate of change and direction of the line. Positive slopes rise from left to right, and negative slopes fall.
• **Y-intercept**: The point where the line crosses the y-axis. It represents the value of \( y \) when \( x = 0 \).
• **Graph**: Linear functions produce straight lines when graphed. The slope and y-intercept together determine the position and orientation of the line.

Linear functions are used extensively in real-world applications to model relationships and predict outcomes, like predicting profits or costs associated with different quantities. Their easy-to-understand nature makes them foundational in algebra and calculus.