Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, use each pair of functions to find \(f(g(x))\) and \(g(f(x))\). Simplify your answers. \(f(x)=|x|, \quad g(x)=5 x+1\)

Step-by-Step Solution

Answer

\(f(g(x)) = |5x + 1|\) and \(g(f(x)) = 5|x| + 1\)

Step 1: Calculate Inner Expression for f(g(x))

Substitute the function \(g(x) = 5x + 1\) into \(f(x) = |x|\). This gives us \(f(g(x)) = |g(x)| = |5x + 1|\). There is no further simplification possible since absolute values depend on the input.

Step 2: Calculate Inner Expression for g(f(x))

Substitute the function \(f(x) = |x|\) into \(g(x) = 5x + 1\). This gives us \(g(f(x)) = 5|x| + 1\). Simplification simply writes as \(g(f(x)) = 5|x| + 1\) with no further steps since we don't have specific values for \(x\).

Key Concepts

Absolute Value Function

Understanding the absolute value function is essential as it frequently appears in algebraic expressions. The absolute value of a number represents its distance from zero on a number line. This means it is always non-negative.
The symbol for absolute value is two vertical bars surrounding the number or expression, like this: \(|x|\).
Here are some important points:

• \(|x| = x\) if \(x > 0\).
• \(|x| = -x\) if \(x < 0\), as you must take the positive of the negative value.
• \(|x| = 0\) when \(x = 0\).

In the exercise, \(f(x) = |x|\), and when calculating \(f(g(x))\), we ended up with \(|5x + 1|\). There is no further simplification possible here unless you know whether \(5x + 1\) is positive or negative.

Piecewise Functions

Piecewise functions are functions defined by different expressions over different parts of their domain. They allow you to describe a function with multiple conditional expressions, making them useful for capturing real-world scenarios.
Think of a piecewise function like having multiple instructions based on different situations. Each piece corresponds to a certain interval of the variable.
Our expression \(f(g(x)) = |5x + 1|\) could be analyzed as a piecewise function:

• If \(5x + 1 >= 0\), \(f(g(x)) = 5x + 1\).
• If \(5x + 1 < 0\), \(f(g(x)) = -(5x + 1)\), reducing to \(-5x - 1\).

Although we often do not specify it in basic exercises, recognizing this aspect helps when analyzing behavior at different points.

Simplifying Expressions

Simplifying expressions involves reducing them into their simplest form while preserving their values. This process often includes combining like terms, factoring, or performing basic arithmetic.
For our functions, simplifying happened during the substitution step. For \(f(g(x))\), we arrived at \(|5x + 1|\) which stays as is unless more context is provided. Conversely, in \(g(f(x))\), substituting results in \(5|x| + 1\).
To simplify effectively:

• Look for elements that can be combined.
• Remember that simplification can sometimes be limited if the expression depends on variable conditions, like absolute value.
• Choose the simplest but most informative version of the expression.

This assists in understanding, especially when dealing with algebraic functions, where concise expression terms can make calculations easier.