Problem 18 - Algebra and Trigonometry

Question

\(17-22=\) Use \(f(x)=3 x-5\) and \(g(x)=2-x^{2}\) to evaluate the expression. $$ \begin{array}{ll}{\text { (a) } f(f(4))} & {\text { (b) } g(g(3))}\end{array} $$

Step-by-Step Solution

Answer

(a) 16, (b) -47

Step 1: Calculate f(4)

To solve for \( f(f(4)) \), we first need to find \( f(4) \). The function \( f(x) \) is given by \( f(x) = 3x - 5 \). Substitute \( x = 4 \) into \( f(x) \): \[ f(4) = 3(4) - 5 = 12 - 5 = 7 \] So, \( f(4) = 7 \).

Step 2: Calculate f(f(4))

Now, we need to calculate \( f(f(4)) \), which is the same as \( f(7) \). We already know that \( f(4) = 7 \), so substitute \( x = 7 \) into \( f(x) \): \[ f(7) = 3(7) - 5 = 21 - 5 = 16 \] Thus, \( f(f(4)) = 16 \).

Step 3: Calculate g(3)

To solve \( g(g(3)) \), we first need to find \( g(3) \). The function \( g(x) \) is given by \( g(x) = 2 - x^2 \). Substitute \( x = 3 \) into \( g(x) \): \[ g(3) = 2 - 3^2 = 2 - 9 = -7 \] So, \( g(3) = -7 \).

Step 4: Calculate g(g(3))

Now, we need to calculate \( g(g(3)) \), which is \( g(-7) \). We already determined that \( g(3) = -7 \), so substitute \( x = -7 \) into \( g(x) \): \[ g(-7) = 2 - (-7)^2 = 2 - 49 = -47 \] Thus, \( g(g(3)) = -47 \).

Key Concepts

Nested Functions

Understanding the concept of nested functions is crucial when solving complex problems in mathematics. A nested function can be thought of as a function within a function.
In simpler terms, you perform one function operation, and then the result becomes the input of another function.For example, in the expression \( f(f(4)) \), first, evaluate \( f(4) \). This gives you a number, which you then plug back into the same function \( f \).
This step-by-step procedure is typical of nested functions, making them versatile tools in mathematical problem-solving.

• Step 1: Evaluate the inner function.
• Step 2: Use the result as the input of the outer function.

This approach helps simplify problems by breaking them into manageable parts, where each nested operation builds on the previous result.

Evaluating Functions

Evaluating functions means calculating the function's output for a specific input value.
This process involves substituting a specific number in place of the variable within the function and simplifying to find the result.When approaching a problem like \( f(f(4)) \), the first step involves finding \( f(4) \).
Here, you substitute \( x = 4 \) into \( f(x) = 3x - 5 \) to get \( f(4) = 3(4) - 5 = 7 \).
You repeat this process for the outer function using the result from the initial evaluation.Useful tips:

• Always follow the function's definition carefully.
• Double-check your arithmetic calculations.
• Substitute accurately, especially with negative and positive values.

Remember, each function's evaluation stands on its own, so accuracy and attention to detail are key to finding the correct result.

Quadratic Functions

Quadratic functions are a type of polynomial function with the highest degree of two, typically in the form \( ax^2 + bx + c \).
In the exercise, the function \( g(x) = 2 - x^2 \) is a quadratic function because the variable \( x \) is raised to the power of two.Understanding how to work with quadratic functions is important because they frequently appear in various mathematical contexts.
Unlike linear functions, which change at a constant rate, quadratic functions produce a curved graph called a parabola.Key attributes:

• The vertex of the parabola represents the minimum or maximum point, depending on the sign of \( a \).
• The function is symmetrical around its vertex.

When evaluating a quadratic function, input a value for \( x \), square it (considering signs), then proceed with the rest of the arithmetic operations as defined by the function.
For instance, \( g(3) = 2 - 3^2 = 2 - 9 = -7 \).
This requires accuracy, especially when dealing with negative results, ensuring each term of the equation is handled properly.