Problem 23 - Algebra and Trigonometry

Question

Use \(f(x)=3 x-5\) and \(g(x)=2-x^{2}\) to evaluate the expression. $$ \begin{array}{ll}{\text { (a) }(f \circ g)(-2)} & {\text { (b) }(g \circ f)(-2)}\end{array} $$

Step-by-Step Solution

Answer

(a) -11, (b) -119

Step 1: Understanding Composition of Functions

When we see notation like \((f \circ g)(x)\), it means we need to evaluate the function \(f\) at the output of the function \(g(x)\). In simpler terms, \((f \circ g)(x) = f(g(x))\). Similarly, \((g \circ f)(x) = g(f(x))\).

Step 2: Calculating \(g(-2)\)

First, we need to evaluate \(g(x)\) at \(x = -2\). The function \(g(x) = 2-x^2\). Thus, \(g(-2) = 2 - (-2)^2 = 2 - 4 = -2\).

Step 3: Calculating \(f(g(-2)) = f(-2)\)

Now that we have \(g(-2) = -2\), we need to find \(f(-2)\). The function \(f(x) = 3x - 5\). So, \(f(-2) = 3(-2) - 5 = -6 - 5 = -11\).

Step 4: Compile Result for \((f \circ g)(-2)\)

We found that \((f \circ g)(-2) = f(g(-2)) = -11\).

Step 5: Calculating \(f(-2)\) Again for \((g \circ f)(-2)\)

Recall \(f(-2)\) calculated as \(-11\) using the previous step.

Step 6: Calculating \(g(f(-2)) = g(-11)\)

Now, evaluate \(g(-11)\) using \(g(x) = 2 - x^2\). Hence, \(g(-11) = 2 - (-11)^2 = 2 - 121 = -119\).

Step 7: Compile Result for \((g \circ f)(-2)\)

Thus, \((g \circ f)(-2) = g(f(-2)) = -119\).

Key Concepts

Function Evaluation

Evaluating a function involves substituting a specific value into the function's formula and carrying out the arithmetic operations. In the context of the exercise, we have two functions, - **Functions**- **\(f(x) = 3x - 5\)** - **\(g(x) = 2 - x^2\)**. When tasked with evaluating \(g(-2)\), you substitute \(-2\) into the function **\(g(x)\)** and perform the calculations:

• Replace \(x\) with \(-2\)
• Calculate \(2 - (-2)^2 = 2 - 4\)
• Which results in \(-2\)

Similarly, for \(f(-2)\), plug \(-2\) into **\(f(x)\)**:

• Substitute \(x = -2\)
• Calculate \(3(-2) - 5 = -6 - 5\)
• The result is \(-11\)

Function evaluation is essential to solve compositions and other complex problems.

Composition Order

Composition of functions, indicated as \((f \, \circ \, g)(x)\), means applying one function to the results of another. It's crucial to perform operations in the correct order. **Order of Composition**- **\((f \, \circ \, g)(x) = f(g(x))\)**- **\((g \, \circ \, f)(x) = g(f(x))\)** For example, to solve \((f \, \circ \, g)(-2)\), first evaluate \(g(-2)\) which gives \(-2\). Then, use this result to find \(f(-2)\) which results in \(-11\). Conversely, for \((g \, \circ \, f)(-2)\), evaluate \(f(-2)\) to get \(-11\), and then compute \(g(-11)\) that results in \(-119\). Ordering is key because reversing the order changes the value of the composition. It exemplifies how functions can transform and build upon each other's outputs.

Algebra Techniques

Using algebra effectively is fundamental in evaluating function compositions. This involves understanding the operations and manipulation of expressions. **Techniques Used**

• **Substitution** - replacing variables with specific numbers or expressions
• **Exponents** - handling powers of numbers, like \((-2)^2 = 4\)
• **Simplification** - combining like terms and reducing expressions

These steps were employed when determining both \(f(g(-2))\) and \(g(f(-2))\). - For example, substituting \(-11\) in **\(g(x)\)** as: - \(2 - (-11)^2 = 2 - 121\)- Applying order of operations to simplify and solve.Mastering these techniques simplifies solving complex problems and strengthens problem-solving skills.