Problem 4 - Precalculus

Question

Use the given pair of functions to find the following values if they exist. $$\bullet (g \circ f)(0)$$ $$\bullet (f \circ g)(-1)$$ $$\bullet (f \circ f)(2)$$ $$\bullet (g \circ f)(-3)$$ $$\bullet (f \circ g)\left(\frac{1}{2}\right)$$ $$\bullet (f \circ f)(-2)$$ $$f(x)=|x-1|, g(x)=x^{2}-5$$

Step-by-Step Solution

Answer

(g ∘ f)(0) = -4, (f ∘ g)(-1) = 5, (f ∘ f)(2) = 0, (g ∘ f)(-3) = 11, (f ∘ g)(1/2) = 23/4, (f ∘ f)(-2) = 2.

Step 1: Understand the Composition of Functions

The composition of functions, such as \((g \circ f)(x)\), means applying function \(f\) first, then applying function \(g\) on the result of \(f(x)\). Similarly, \((f \circ g)(x)\) means applying \(g\) first, then \(f\).

Step 2: Evaluate (g ∘ f)(0)

First, evaluate \(f(0)\):\[ f(0) = |0 - 1| = 1 \]Then evaluate \(g(f(0))\):\[ g(1) = 1^2 - 5 = -4 \]Thus, \((g \circ f)(0) = -4\).

Step 3: Evaluate (f ∘ g)(-1)

First, evaluate \(g(-1)\):\[ g(-1) = (-1)^2 - 5 = 1 - 5 = -4 \]Then evaluate \(f(g(-1))\):\[ f(-4) = |-4 - 1| = 5 \]Thus, \((f \circ g)(-1) = 5\).

Step 4: Evaluate (f ∘ f)(2)

First, evaluate \(f(2)\):\[ f(2) = |2 - 1| = 1 \]Then evaluate \(f(f(2))\):\[ f(1) = |1 - 1| = 0 \]Thus, \((f \circ f)(2) = 0\).

Step 5: Evaluate (g ∘ f)(-3)

First, evaluate \(f(-3)\):\[ f(-3) = |-3 - 1| = 4 \]Then evaluate \(g(f(-3))\):\[ g(4) = 4^2 - 5 = 16 - 5 = 11 \]Thus, \((g \circ f)(-3) = 11\).

Step 6: Evaluate (f ∘ g)(1/2)

First, evaluate \(g(1/2)\):\[ g\left( \frac{1}{2} \right) = \left( \frac{1}{2} \right)^2 - 5 = \frac{1}{4} - 5 = -\frac{19}{4} \]Then evaluate \(f(g(1/2))\):\[ f\left( -\frac{19}{4} \right) = \left| -\frac{19}{4} - 1 \right| = \left| -\frac{23}{4} \right| = \frac{23}{4} \]Thus, \((f \circ g)\left( \frac{1}{2} \right) = \frac{23}{4}\).

Step 7: Evaluate (f ∘ f)(-2)

First, evaluate \(f(-2)\):\[ f(-2) = |-2 - 1| = 3 \]Then evaluate \(f(f(-2))\):\[ f(3) = |3 - 1| = 2 \]Thus, \((f \circ f)(-2) = 2\).

Key Concepts

Absolute Value Functions

An absolute value function is a mathematical function that reflects how far a number is from zero on the number line, without considering direction. This means it always results in a non-negative value. The absolute value of a number is denoted by two vertical bars, for example, \( |x| \). This operation essentially strips away any negative sign that might be present. Let's look at how it works:

• The absolute value of a positive number remains unchanged; for example, \( |5| = 5 \).
• The absolute value of a negative number becomes positive; for example, \( |-5| = 5 \).
• The absolute value of zero is zero; \( |0| = 0 \).

In the context of the exercise, the function \( f(x) = |x - 1| \) takes any input value \( x \), subtracts one, and then changes it to its absolute value. This means the result is always non-negative regardless of whether \( x - 1 \) yields a positive or negative number.

Quadratic Functions

Quadratic functions are a type of polynomial function with the highest exponent being two. They are expressed in the form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants, and \( a \) is not zero. These functions create a characteristic 'U'-shaped graph known as a parabola. Key features to consider include:

• The vertex of the parabola, which is the highest or lowest point depending on the orientation.
• The axis of symmetry, a vertical line that passes through the vertex dividing the parabola into two mirrored halves.
• The direction of the parabola, up or down, is determined by the coefficient \( a \). If \( a \) is positive, it opens upwards, and if negative, downwards.

In this exercise, the quadratic function given is \( g(x) = x^2 - 5 \). Here, the vertex is at the central point of this expression, and since the leading term \( x^2 \) is positive, the parabola opens upward, indicating that the function will have a minimum point rather than a maximum.

Function Evaluation

Function evaluation involves substituting a variable within a function with a specific value to find the output. This process is crucial for understanding what a function yields when applied to different inputs. For instance, given a function \( f(x) \,\) evaluating \( f(a) \) means calculating the value of the function after replacing \( x \) with \( a \).
In the context of this problem:

• To find \( (g \circ f)(0) \), we first evaluate \( f(0) \) and then use that result to find \( g(f(0)) \).
• Similarly, finding \( (f \circ f)(-2) \) involves evaluating \( f(-2) \) first and using that result for another evaluation of \( f \).

Function evaluation helps us understand how compositions such as \( (g \circ f) \) or \( (f \circ g) \) are resolved to get a specific output.

Mathematical Operations

Mathematical operations are the building blocks used to construct and evaluate expressions and functions. They include basic operations such as addition, subtraction, multiplication, and division. Each of these operations can change how values are manipulated and combined.
In the current exercise, we see these operations working hand in hand to evaluate composite functions:

• Subtraction is used when finding the absolute value, as in \( |x - 1| \).
• Exponents are a key mathematical tool used in quadratic functions like \( x^2 \). This operation moves values to a second power, introducing a parabolic behavior.
• Addition or subtraction follows when shifting or adjusting the results, visible in the \( -5 \) of \( g(x) = x^2 - 5 \).

Understanding these operations is critical because each plays a role in determining the function's output during evaluation.