Problem 28 - Algebra for College Students

Question

If \(f(x)=x^{2}-2\) and \(g(x)=x+4\), find \((f \circ g)(2)\) and \((g \circ f)(-4) . \quad 34 ; 18\)

Step-by-Step Solution

Answer

The values are 34 and 18.

Step 1: Understand Composition

When we see notation like \((f \circ g)(x)\), it means we need to substitute \(g(x)\) into the function \(f(x)\). Similarly, \((g \circ f)(x)\) means substituting \(f(x)\) into \(g(x)\). We will compute these compositions for specific values of \(x\).

Step 2: Evaluate \(g(2)\) for Composition

To find \((f \circ g)(2)\), we first need to evaluate \(g(2)\). Substituting 2 into \(g(x) = x + 4\), we get:\[ g(2) = 2 + 4 = 6 \]

Step 3: Evaluate \(f(g(2))\)

Now that we know \(g(2) = 6\), we substitute 6 into \(f(x) = x^2 - 2\):\[ f(g(2)) = f(6) = 6^2 - 2 = 36 - 2 = 34 \]

Step 4: Evaluate \(f(-4)\) for Composition

To find \((g \circ f)(-4)\), we first evaluate \(f(-4)\). Substituting -4 into \(f(x) = x^2 - 2\), we have:\[ f(-4) = (-4)^2 - 2 = 16 - 2 = 14 \]

Step 5: Evaluate \(g(f(-4))\)

Now that we know \(f(-4) = 14\), we substitute 14 into \(g(x) = x + 4\):\[ g(f(-4)) = g(14) = 14 + 4 = 18 \]

Key Concepts

Composite Functions

Composite functions combine two or more functions into a single operation. This is denoted by

• \((f \circ g)(x)\), meaning "\(f\) composed with \(g\)" or "apply \(g\) first, then \(f\)".
• \((g \circ f)(x)\), meaning "\(g\) composed with \(f\)" or "apply \(f\) first, then \(g\)".

Understanding this concept is like following a set of instructions where each step depends on the result of the previous. In our exercise,

• To find \((f \circ g)(2)\), we substitute 2 into \(g(x)\) first to get a new value, then substitute this result into \(f(x)\).
• Similarly, to find \((g \circ f)(-4)\), we first evaluate \(f(-4)\) and use this output as the input for \(g(x)\).

This simple back-and-forth process helps in solving complex problems in a structured way.

Function Evaluation

Function evaluation involves substituting specific values into a function's formula to find the corresponding output. Consider the definition of each function:

• For \(f(x) = x^2 - 2\), substituting a value for \(x\) will give you a result by squaring \(x\) and subtracting 2.
• For \(g(x) = x + 4\), the operation is adding 4 to the value of \(x\).

In our exercise, when solving for \((f \circ g)(2)\), you start with \(g(2)\), meaning substitute 2 into \(g\). This gives \( g(2) = 6 \).
Next, plug this result into \(f(x)\): \(f(g(2)) = f(6) = 6^2 - 2\). Continue this step-by-step process for different compositions to ensure accuracy and solve for both \((f \circ g)(2)\) and \((g \circ f)(-4)\).Using specific numerical values not only helps in computations but also supports our understanding of how functions transform inputs into outputs.

Algebraic Functions

Algebraic functions, like those in the exercise, express relationships between variables using basic operations: addition, subtraction, multiplication, division, and more. The idea of algebraic functions allows us to handle unknown quantities and express real-world situations symbolically.
In our scenario:

• \(f(x)=x^{2}-2\) is a quadratic function formed by squaring \(x\) and subtracting 2.
• \(g(x)=x+4\) is a linear function, simply adding 4 to \(x\).

Dealing with algebraic functions requires understanding how these operations affect the input values and transform them according to given rules.
This enables a clearer comprehension of how different functions interact through operations such as composition, where results from one function serve as inputs for another. Such functions form the backbone of algebra, enabling problem-solving and the exploration of more complex mathematical concepts.