Problem 28 - Precalculus with Limits

Question

In Exercises 17-28, evaluate the indicated function for \(f(x) = x^2 + 1\) and \(g(x) = x - 4\). \((fg)(5)+f(4)\)

Step-by-Step Solution

Answer

The value of the given expression is 19.

Step 1: Determine fg(5)

Calculate \(f(g(5))\) or \(f(g(x))\) at x=5. To do so, first solve for \(g(5)\) which equals \((5-4) = 1\). Plug the result into f(x) and solve for \(f(1)\), which gives \((1)^2 + 1 = 2\). Therefore, \(f(g(5)) = 2\).

Step 2: Evaluate f(4)

Solve for \(f(4)\). Plug 4 into f(x) which gives \(4^2 + 1 = 17\). Therefore, \(f(4) = 17\).

Step 3: Evaluate the final expression

Lastly, add together the two results to evaluate the whole expression, \((fg)(5) + f(4) = 2 + 17 = 19\).

Key Concepts

Function Evaluation

When we talk about function evaluation, we are referring to the process of determining the output of a function given a specific input. In mathematical terms, if you have a function expressed as \( f(x) \), to evaluate this function means to substitute a chosen number from the function's domain in place of \( x \) and perform the operations defined by the function to find the result.
This practice becomes particularly interesting when evaluating composite functions. A composite function like \( f(g(x)) \) involves finding the result of one function, \( g(x) \), and then using that result as the input for another function, \( f(x) \).

• First, solve \( g(x) \) for your specific input. For example, find \( g(5) \) by plugging 5 into \( g(x) = x - 4 \), resulting in 1.
• Then, use this result in the function \( f(x) \). So, you find \( f(1) \) by plugging 1 into \( f(x) = x^2 + 1 \), resulting in 2.

This step-by-step process illustrates the procedural chain of function evaluations and highlights their foundational importance in mathematics.

Quadratic Functions

Quadratic functions are a fundamental category of polynomial functions represented in the general form \( f(x) = ax^2 + bx + c \). They are known for their characteristic U-shaped graphs called parabolas.
In the given exercise, we have a specific quadratic function \( f(x) = x^2 + 1 \). Here are some basic features of this function:

• **Vertex**: Since \( f(x) = x^2 + 1 \) lacks a \( b \) term, the parabola is symmetrical around the vertical line \( x = 0 \), and its vertex is at \( (0, 1) \).
• **Direction**: The coefficient \( a \) is positive (\( a = 1 \)), indicating the parabola opens upwards.
• **Y-intercept**: With \( c = 1 \), the curve crosses the \( y \)-axis at \( (0, 1) \).
• **Domain and Range**: The domain of all quadratic functions is all real numbers. For our \( f(x) \), the range starts at \( y = 1 \) and goes to infinity.

Understanding these aspects of quadratic functions is essential in solving problems involving them and analyzing their behaviors.

Arithmetic Operations

Arithmetic operations refer to basic computations such as addition, subtraction, multiplication, and division. In the context of function evaluation, these involve manipulating the values obtained from evaluated functions to reach a final result.
In this exercise, you're tasked with finding \((fg)(5) + f(4)\). This involves a few arithmetic steps:

• First, find \( f(g(5)) \). You already know this equals 2.
• Next, determine \( f(4) \), which equals 17.
• Finally, perform the addition: \( 2 + 17 = 19 \).

Every arithmetic operation performed here needs to follow the conventional order of operations (also known as PEMDAS/BODMAS), ensuring that calculations are performed in a logical and mathematically sound manner. By correctly evaluating, we combine function insights with arithmetic skills to find accurate results.