Problem 117 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the function at each specified value of the independent variable and simplify. \(f(x)=-x^{2}-x+3\) (a) \(f(4)\) (b) \(f(-5)\) (c) \(f(x-2)\)

Step-by-Step Solution

Answer

The evaluated function at \(x = 4\) and \(x = -5\) is \(-17\). The simplified function form for \(x = x - 2\) is \(-x^{2}+4x-1\).

Step 1: Evaluate the function at \(x = 4\)

Replace \(x\) with \(4\) in the given equation. \(f(4)=-(4)^{2}-4+3 = -16-4+3 = -17\)

Step 2: Evaluate the function at \(x = -5\)

Replace \(x\) with \(-5\) in the given equation. \(f(-5)=-(-5)^{2}-(-5)+3 = -25+5+3 = -17\)

Step 3: Evaluate the function at \(x = x - 2\)

Replace \(x\) with \(x-2\) in the given equation. \(f(x-2)=-(x-2)^{2}-(x-2)+3 = -(x^{2}-4x+4)-(x-2)+3 = -x^{2}+4x-1\)

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in algebra, allowing us to find the output value of a function given a specific input. Think of functions as machines. You put a specific value in (like a number or an expression), and the function processes it to give you an output.
To evaluate a function, such as our example function, \( f(x) = -x^2 - x + 3 \), we replace the variable \( x \) with the given input value. Let's break it down with simple steps to ensure you grasp the process:

• Identify the input value you need to evaluate, such as \( x = 4 \), \( x = -5 \), or any expression like \( x-2 \).
• Substitute this value directly into the function's equation to replace the variable \( x \).
• Carry out the arithmetic operations meticulously.

This method helps uncover the output of the function for any given input, providing crucial insights into the behavior of algebraic functions.

Quadratic Functions

Quadratic functions are a central topic in algebra, identifiable by their characteristic \( x^2 \) term. These functions generally have the form \( ax^2 + bx + c \). Our example, \( f(x) = -x^2 - x + 3 \), fits this pattern with \( a = -1 \), \( b = -1 \), and \( c = 3 \).
Quadratic functions often graph as parabolas, which can open upwards or downwards depending on the sign of \( a \). In our example, since \( a = -1 \), the parabola opens downwards.
Understanding the role of each coefficient in a quadratic function is important:

• The "\( a \)" coefficient affects the parabola's width and direction.
• The "\( b \)" coefficient shifts the parabola along the x-axis.
• The "\( c \)" term shifts the parabola up or down the y-axis.

When evaluating a quadratic function, especially in the context of our previous examples, you're essentially predicting where on this parabola the function will fall for given values or expressions of \( x \). It's like pinpointing spots along the curve.

Simplifying Expressions

Simplifying expressions is a technique we use to make mathematical expressions more manageable. When evaluating functions like \( f(x) = -x^2 - x + 3 \), simplifying can streamline complex expressions, making them easier to understand and solve.
For example, in the process of evaluating \( f(x-2) \), we substituted \( x \) with \( x-2 \) and then followed simplification steps:

• Expand any squared or complex expressions, such as \((x-2)^2 = x^2 - 4x + 4\).
• Carefully combine like terms.
• Simplify the expression by performing arithmetic operations.

This process is similar to tidying up a room: you put everything in its place, remove unnecessary items (like excess parentheses or terms), and end up with a clean, easy-to-read expression. Simplification is the key to making algebra less overwhelming and more intuitive.