Problem 68 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

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

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

To find \(f(a)\), substitute \(x\) with \(a\) in the function \(f(x)=x^2\). This gives \(f(a) = a^2\).

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

For \(f(b+1)\), replace \(x\) with \(b+1\) in \(f(x)=x^2\). Using this, we derive \(f(b+1) = (b+1)^2 = b^2 + 2b + 1\).

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

Substitute \(x\) with \(3x\) in \(f(x)=x^2\) to find \(f(3x)\). It results in \(f(3x) = (3x)^2 = 9x^2\).

Key Concepts

Substitution Method

The substitution method is a fundamental technique in algebra used to evaluate functions. This involves replacing a variable with a specific value or expression. For example, if you have a function like \[ f(x) = x^2 \]and you're asked to find \( f(a) \), you would simply substitute \( a \) for \( x \). This gives us

• \( f(a) = a^2 \)
• \( f(b+1) = (b+1)^2 = b^2 + 2b + 1 \)
• \( f(3x) = (3x)^2 = 9x^2 \)

This method is not only useful for evaluating specific values in algebraic expressions, but also in broader mathematical applications where functions need to be analyzed at specific points or with specific input values. The substitution method is a critical stepping stone to understanding more complex algebraic concepts, as it allows you to move variables around and see how changes affect outputs.

Quadratic Function

A quadratic function is a type of polynomial function where the highest degree of the variable is two. In simple terms, it can be written in the standard form:\[ f(x) = ax^2 + bx + c \]where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). The graph of a quadratic function is a parabola. It can open upwards or downwards depending on the sign of \( a \).In our function \( f(x) = x^2 \), the equation is a simplified quadratic function where \( a = 1 \) and both \( b \) and \( c \) are equal to zero. Here are some important characteristics of quadratic functions:

• Vertex: The point of the parabola that is the highest or lowest.
• Axis of Symmetry: A vertical line that divides the parabola into two mirror images.
• Roots: Also known as zeros, these are the values of \( x \) where \( f(x) = 0 \).

Quadratic functions are foundational in algebra and appear frequently in equations modeling real-world scenarios such as physics or economics.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (such as addition or multiplication). They represent mathematical phrases and can be as simple as a single number or more complex like our example from step-by-step solution:\[ f(b+1) = b^2 + 2b + 1 \]This specific expression emerged from evaluating the quadratic function \( f(x) = x^2 \) by replacing \( x \) with \( b+1 \). It's handled through the substitution method and expanded using algebraic rules like the distributive property.Key elements of algebraic expressions include:

• Constants: Standalone numbers.
• Variables: Letters representing unknown values.
• Coefficients: Numbers that multiply the variables.
• Terms: Individual parts of an expression separated by plus or minus signs.

Understanding how to manipulate and simplify these expressions is vital, as they form the backbone of solving equations, modeling phenomena with functions, and conducting algebraic operations.