Problem 6 - Intermediate Algebra

Question

Determine \((f \circ g)(x)\) and \((g \circ f)(x)\) for each pair of functions. Also specify the domain of \((f \circ g)(x)\) and \((g \circ f)(x)\). (Objective 1\()\) \(f(x)=-2 x+4\) and \(g(x)=2 x^{2}-1\)

Step-by-Step Solution

Answer

\((f \circ g)(x) = -4x^2 + 6\); \((g \circ f)(x) = 8x^2 - 32x + 31\). Both have the domain of all real numbers.

Step 1: Understand Composition of Functions

The composition of functions \((f \circ g)(x)\) means that we substitute \(g(x)\) into \(f(x)\). This results in replacing \(x\) in \(f(x)\) with \(g(x)\). Similarly, \((g \circ f)(x)\)does the reverse, substituting \(f(x)\) into \(g(x)\).

Step 2: Calculate \((f \circ g)(x)\)

We know \(f(x) = -2x + 4\) and \(g(x) = 2x^2 - 1\).To find \((f \circ g)(x)\), we substitute \(g(x)\) into \(f(x)\):\[f(g(x)) = f(2x^2 - 1) = -2(2x^2 - 1) + 4\]\[= -4x^2 + 2 + 4\]\[= -4x^2 + 6\]

Step 3: Calculate \((g \circ f)(x)\)

Now substitute \(f(x)\) into \(g(x)\):\[g(f(x)) = g(-2x + 4) = 2(-2x + 4)^2 - 1\]First, simplify \((-2x + 4)^2 = (4x^2 - 16x + 16)\).Then substitute back:\[= 2(4x^2 - 16x + 16) - 1\]\[= 8x^2 - 32x + 32 - 1\]\[= 8x^2 - 32x + 31\]

Step 4: Determine Domain for \((f \circ g)(x)\) and \((g \circ f)(x)\)

The domain of \(g(x) = 2x^2 - 1\) is all real numbers since it is a quadratic function and defined for every real \(x\).The function \(f(x) = -2x + 4\) is also defined for all real \(x\).Thus, the domains for both \((f \circ g)(x) = -4x^2 + 6\) and \((g \circ f)(x) = 8x^2 - 32x + 31\) are all real numbers.

Key Concepts

Domain of a Function

When working with functions, understanding the domain is crucial. The domain of a function is essentially the set of all possible input values (often denoted as \(x\)) that will produce a valid output. To determine the domain, observe any restrictions that could arise when substituting values into the function.

For most functions expressed using simple algebraic expressions, such as polynomials or linear functions, the domain includes all real numbers. This means you can plug in any real number for \(x\), and the function will return a valid output.

However, for functions involving fractions or square roots, special care needs to be taken to avoid division by zero or taking the square root of a negative number. Hence, always look for steps that might restrict the input values, as these affect the overall domain of the function.

• Polynomials: Domain is all real numbers.
• Square roots: Domain is restricted to non-negative numbers.
• Fractions: Domain excludes values making the denominator zero.

Quadratic Functions

Quadratic functions are a type of polynomial function characterized by the expression \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). Their graph forms a parabola, which can open upwards or downwards depending on the sign of the leading coefficient \(a\).

For example, in the function \(g(x) = 2x^2 - 1\), the coefficient \(a = 2\), which is positive. This indicates the parabola opens upwards. Quadratic functions are particularly interesting because they always provide a domain of all real numbers. This is because the \(x^2\) term can handle both positive and negative inputs equally.

• The vertex of the parabola gives the maximum or minimum point.
• Opening direction shows based on the sign of \(a\).
• The axis of symmetry is vertical and passes through the vertex.

Linear Functions

Linear functions have the general form \(f(x) = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. These functions are called 'linear' because their graph is a straight line.

For instance, consider \(f(x) = -2x + 4\). Here, the slope \(m = -2\) indicates that the line is descending as it moves to the right. The y-intercept \(b = 4\) marks the point where the line crosses the y-axis. Linear functions are defined for all real numbers, suggesting their domain spans across all possible real values.

• Slope \(m\) determines the steepness and direction of the line.
• Y-intercept \(b\) specifies where the line crosses the y-axis.
• Constant rate of change makes them easy to model real-world situations.