Problem 32 - Beginning and Intermediate Algebra

Question

Let \(f(x)=3 x-7\) and \(g(x)=x^{2}-4 x-9 .\) Find each of the following and simplify. $$f(z)$$

Step-by-Step Solution

Answer

\(f(z) = 3z - 7\)

Step 1: Substitute \(x\) with \(z\) in \(f(x)\)

In order to find \(f(z)\), we will replace \(x\) with \(z\) in the given function \(f(x)\): \[f(z) = 3z - 7\]

Step 2: Simplify the expression

Since there are no further operations to perform, the expression is already in its simplest form: \[f(z) = 3z - 7\] The function \(f(z)\) is equal to \(3z-7\).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the output of a function given an input. In simple terms, it means plugging in a particular value into the function and simplifying to find the result. When dealing with functions like \( f(x) \), we substitute \( x \) with the value we want to evaluate. For instance, in a function \( f(x) = 3x - 7 \), if we are asked to find \( f(z) \), we change the \( x \) to \( z \). The process involves replacing every instance of the variable \( x \) with \( z \). This is a fundamental skill in algebra, as it allows us to understand how functions behave with different inputs. Remember, evaluating functions can include substituting in numbers, other variables, or even complex expressions.

Substitution Method

The substitution method is a crucial tool when working with algebraic functions. It involves replacing one variable with another value or expression to simplify our calculations.

• Step-by-step, you identify the variable in the function you want to replace.
• Then, you substitute this variable with a given value or another variable.

For example, given \( f(x) = 3x - 7 \), substituting \( x \) with \( z \) involves writing \( 3z - 7 \). It allows mathematicians to transform and solve equations by systematically replacing terms. Using substitution, we can evaluate functions at specific points, such as \( f(z) \), or even solve systems of equations.

Simplification

Simplification in mathematics involves reducing an equation or expression to its most concise form. This means removing any unnecessary terms or combining like terms to make the expression easier to work with.In algebra, once variables are substituted into a function, simplify if there are any terms to combine or coefficients to distribute. For the function \( f(z) = 3z - 7 \), there are no terms that can be combined or further operations required, so it is already simplified.Simplification makes it easier to interpret the results of evaluations and calculations. This step is vital to ensure accuracy and clarity in mathematics, especially when extended calculations are involved.

Linear Function

A linear function is a type of algebraic function where the graph is a straight line. These functions have the general form \( f(x) = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.

• The slope \( m \) indicates how steep the line is.
• The y-intercept \( b \) is the point where the line crosses the y-axis.

The function \( f(x) = 3x - 7 \) is a linear function. In this case:

• The slope \( m = 3 \) tells us that for every unit increase in \( x \), \( f(x) \) increases by 3.
• The y-intercept \( b = -7 \) means the line crosses the y-axis at \(-7\).

Linear functions like this are simple yet powerful for modeling relationships where one variable changes at a constant rate with respect to another.