Problem 69 - A Graphical Approach to College Algebra

Question

Find \(f(a), f(b+1),\) and \(f(3 x)\) for the given \(f(x)\) $$f(x)=1-x^{2}$$

Step-by-Step Solution

Answer

\( f(a) = 1 - a^2 \); \( f(b+1) = -b^2 - 2b \); \( f(3x) = 1 - 9x^2 \).

Step 1: Find \( f(a) \)

To find \( f(a) \), substitute \( a \) into the function \( f(x) = 1 - x^2 \). This gives \( f(a) = 1 - a^2 \).

Step 2: Find \( f(b+1) \)

To find \( f(b+1) \), substitute \( b+1 \) into the function \( f(x) = 1 - x^2 \). This gives \( f(b+1) = 1 - (b+1)^2 \). Simplify further: \[ f(b+1) = 1 - (b^2 + 2b + 1) = 1 - b^2 - 2b - 1 = -b^2 - 2b \]

Step 3: Find \( f(3x) \)

To find \( f(3x) \), substitute \( 3x \) into the function \( f(x) = 1 - x^2 \). This gives \( f(3x) = 1 - (3x)^2 \). Simplify further:\[ f(3x) = 1 - 9x^2 \]

Key Concepts

Quadratic Functions

Quadratic functions are a crucial part of algebra and appear in many mathematical contexts. They are functions of the form

• \( f(x) = ax^2 + bx + c \)

where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In the given exercise, the function \( f(x) = 1 - x^2 \) is a quadratic function, where the coefficient \( a = -1 \), and both \( b \) and \( c \) are 0 and 1 respectively.

The graph of a quadratic function is a parabola. Depending on the value of \( a \), it can open either upwards or downwards. If \( a \) is positive, the parabola opens upwards, and if \( a \) is negative, as in our function, it opens downwards.

Quadratic functions often appear in problems related to motion, physics, business optimization, and many others, making them one of the most versatile and important classes of functions in mathematics.

Function Evaluation

Evaluating a function means finding its value at a specific point. This involves substituting a given value of \( x \) into the function and calculating the result. Let's dive into how this process works for the given quadratic function:

- For \( f(a) \), replace \( x \) with \( a \) in the function \( f(x) = 1 - x^2 \), giving us \( f(a) = 1 - a^2 \).- For \( f(b+1) \), substitute \( b+1 \) into the same function, leading to \( f(b+1) = 1 - (b+1)^2 \). After expanding and simplifying, this becomes \( -b^2 - 2b \).- For \( f(3x) \), substitute \( 3x \) in place of \( x \) in the function, yielding \( f(3x) = 1 - 9x^2 \).

Understanding function evaluation is essential because it allows you to find specific outputs for given inputs, which can help solve real-world problems or optimize various systems and models.

Substitution Method

The substitution method is a mathematical tool used to simplify and solve problems involving functions, especially when you need to evaluate or manipulate expressions. Here's how it applies to the given problem:

When you substitute a value or an expression into a function, you replace every instance of the variable with the given number or expression.

• For \( f(a) \), you replace \( x \) with \( a \), resulting in the expression \( 1 - a^2 \).
• Substituting \( b+1 \) into \( f(x) \) involves more steps. You need to substitute \( b+1 \) for \( x \), expand \( (b+1)^2 \) to \( b^2 + 2b + 1 \), and finally simplify the expression to \( -b^2 - 2b \).
• When substituting \( 3x \), replace \( x \) with \( 3x \) and simplify \((3x)^2\) to get \( 9x^2 \), resulting in \( 1 - 9x^2 \).

This method allows for clear and structured manipulation of functions, making it easier to handle complex expressions and ensuring accurate results in calculations.