Problem 27 - College Algebra

Question

For the following exercises, evaluate the function \(f\) at the indicated values \(f(-3), f(2), f(-a),-f(a), f(a+h)\). $$ f(x)=2 x-5 $$

Step-by-Step Solution

Answer

\(f(-3) = -11\), \(f(2) = -1\), \(f(-a) = -2a - 5\), \(-f(a) = -2a + 5\), \(f(a+h) = 2a + 2h - 5\).

Step 1: Evaluate f(-3)

To find \(f(-3)\), substitute \(-3\) into the function for \(x\). The function is \(f(x) = 2x - 5\). Thus, \(f(-3) = 2(-3) - 5 = -6 - 5 = -11\).

Step 2: Evaluate f(2)

Substitute \(2\) into the function for \(x\). Using the function \(f(x) = 2x - 5\), we get \(f(2) = 2(2) - 5 = 4 - 5 = -1\).

Step 3: Evaluate f(-a)

Replace \(x\) with \(-a\) in the function. Thus, \(f(-a) = 2(-a) - 5 = -2a - 5\).

Step 4: Evaluate -f(a)

Find \(f(a)\) first, which is \(f(a) = 2a - 5\). Then, multiply by \(-1\), so \(-f(a) = -(2a - 5) = -2a + 5\).

Step 5: Evaluate f(a+h)

Substitute \(a+h\) for \(x\) in the function. \(f(a+h) = 2(a+h) - 5 = 2a + 2h - 5\).

Key Concepts

Substitution

Substitution is a fundamental method used in mathematics, especially when working with functions. It involves replacing a variable in an expression with a specific value. Substitution helps to evaluate the function at various points. This process allows us to simplify expressions and get a numerical or algebraic answer depending on the context.

For example, in the function given as \( f(x) = 2x - 5 \), if we want to find \( f(-3) \), we substitute \(-3\) for \(x\). This step involves replacing every instance of \(x\) in the expression with \(-3\) which gives us \( 2(-3) - 5 \). Simplifying leads to the result: \(-11\).

Substitution is not limited to numbers alone. Variables can also be substituted. For instance, replacing \(x\) with \(a + h\), which means wherever \(x\) appears in the function, it gets replaced with \(a + h\). This allows us to explore how the function behaves with varied inputs.

Algebraic Expressions

Algebraic expressions are mathematical phrases that contain numbers, variables, and operational symbols, such as addition or subtraction.

In our example, the function \( f(x) = 2x - 5 \) is an algebraic expression. It describes a particular relationship between the input \(x\) and the resulting output \(f(x)\). Here, \(2x\) represents the product of \(2\) and \(x\), and \(-5\) indicates subtraction of \(5\) from \(2x\).

When replacing parts of an algebraic expression, the rules of algebra still apply. For instance, substituting \(-a\) for \(x\) in the function yields \( 2(-a) - 5 \) which simplifies to \(-2a - 5\). The operations of multiplication and subtraction here follow the same process, but now with a variable instead of a number, demonstrating that algebraic expressions are dynamic and can be manipulated to understand different contexts or inputs.

Linear Functions

Linear functions are a kind of function that creates a straight line when plotted on a graph. They are characterized by expressions of the form \( f(x) = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept.

In our case, \( f(x) = 2x - 5 \) is a linear function, where \(2\) is the slope (\(m\)) and \(-5\) is the y-intercept (\(b\)). The slope tells us how steep or flat the line is when plotted, while the y-intercept indicates where the line crosses the y-axis.

Evaluating linear functions at different points, like finding \( f(a) \) or \( -f(a) \), provides further instances of how the function behaves with different kinds of inputs. Each evaluation changes the point on the line but maintains the linear relationship defined by the function. This property makes linear functions very predictable and a useful tool in modeling relationships in various fields.