Problem 50 - Beginning and Intermediate Algebra

Question

Let \(f(x)=-5 x+2\) and \(g(x)=x^{2}+7 x+2 .\) Find each of the following and simplify. $$f(t-6)$$

Step-by-Step Solution

Answer

The short answer is: \(f(t-6) = -5t + 32\).

Step 1: Substitute (t-6) for x in f(x)

Replace \(x\) with \((t-6)\) in the function \(f(x)=-5x+2\): $$f(t-6) = -5(t-6) + 2$$

Step 2: Simplify the expression

Now, we'll simplify the expression by distributing the \(-5\) and combining like terms: $$f(t-6) = -5t + 30 + 2$$ $$f(t-6) = -5t + 32$$ Therefore, the expression for \(f(t-6)\) is: $$f(t-6) = -5t + 32$$

Key Concepts

Substitute Variable

The concept of substituting a variable is fundamental in algebra and calculus, particularly when working with functions. When you substitute a variable, you're essentially taking the place of the variable in the function with another expression or value. This can help evaluate or transform the function to suit your needs. In the given problem, we have the function \( f(x) = -5x + 2 \).

• You're asked to find \( f(t-6) \). This means we replace \( x \) in the function with \( (t-6) \).
• By substituting, you transform \( f(x) \) into \( f(t-6) = -5(t-6) + 2 \).

Substituting variables allows you to study how functions behave under different circumstances by simply replacing variables with other terms or values, thus opening doors to solve more complex problems.

Simplify Expression

Simplifying expressions is an essential algebraic skill. It involves combining like terms and applying distributive properties to create a shorter and more manageable version of an algebraic expression. After substituting \( (t-6) \) into \( f(x) = -5x + 2 \), it's necessary to simplify to find \( f(t-6) \).First, you distribute the \( -5 \) over the terms in the parenthesis:

• The multiplication \( -5(t-6) \) results in \( -5t + 30 \).
• Adding the constant term \( 2 \) gives us \( -5t + 30 + 2 \).
• Finally, by combining like terms, we simplify it to \( -5t + 32 \).

Simplifying expressions involves breaking down each part, then reassembling them in a straightforward manner.

Algebraic Functions

Algebraic functions are mathematical expressions built using algebraic operations such as addition, subtraction, multiplication, division, and exponentiation with constants and variables. In the exercise given, two functions are defined:

• \( f(x) = -5x + 2 \), a linear function.
• \( g(x) = x^2 + 7x + 2 \), a quadratic function.

Functions like these are prevalent in mathematical analysis because they describe a wide variety of phenomena. Understanding these functions involves recognizing how the operation on a variable affects its output and how substituting different values or expressions for the variable can yield useful insights into the function's behavior. This is why substituting \( t-6 \) into \( f(x) \) alters its standard interaction, changing the output accordingly. Algebraic functions provide a fundamental language for describing mathematical relationships.