Problem 6
Question
Evaluate the function \(f(x)=-5 x-2\) for the given value of \(x\) $$ x=-\frac{1}{5} $$
Step-by-Step Solution
Verified Answer
The function \(f(-\frac{1}{5}) = -1\).
1Step 1: Substitute the Given Value into the Function
Substitute \(x = -\frac{1}{5}\) into the function \(f(x) = -5x - 2\). That will result in \(f(-\frac{1}{5}) = -5(-\frac{1}{5}) - 2\).
2Step 2: Simplify the Function with the Value
Simplify: \(f(-\frac{1}{5}) = 1 - 2\).
3Step 3: Calculate the Result
After calculating the operation, we get \(f(-\frac{1}{5}) = -1\).
Key Concepts
Function EvaluationSubstitution Method in AlgebraAlgebraic Expressions Simplification
Function Evaluation
Function evaluation is akin to following a recipe to bake a specific cake: the recipe provides the process, and the ingredients you choose to use, like the value of 'x' in algebra, determine the final taste of your creation. In mathematics, functions are recipes that tell you how to combine inputs to get a desired output. For example, if we have a function like \(f(x) = -5x - 2\), it means that for any number we choose for 'x', we should multiply it by -5 and then subtract 2 to find our output.
When a specific value is provided, as in our exercise where \(x=-\frac{1}{5}\), we evaluate the function by substituting this value in place of 'x'. This process transforms the algebraic expression into a numerical one which we can compute for a concrete result. Remember, precise substitution is critical to obtaining the correct evaluation of the function.
When a specific value is provided, as in our exercise where \(x=-\frac{1}{5}\), we evaluate the function by substituting this value in place of 'x'. This process transforms the algebraic expression into a numerical one which we can compute for a concrete result. Remember, precise substitution is critical to obtaining the correct evaluation of the function.
Substitution Method in Algebra
The substitution method in algebra is a fundamental tool that allows us to find the value of algebraic expressions with given variables. Think of it as a 'find and replace' feature. You are given a letter that represents a number, and you must replace that letter with the numerical value it corresponds to.
For example, with our function \(f(x) = -5x - 2\), when we're told that \(x = -\frac{1}{5}\), we substitute every 'x' in the expression with '-\frac{1}{5}'. Doing this correctly is crucial; even a small mistake can throw off the entire solution. After replacing 'x' with its value, the expression should be simplified, which leads us into our next concept, the simplification of algebraic expressions.
For example, with our function \(f(x) = -5x - 2\), when we're told that \(x = -\frac{1}{5}\), we substitute every 'x' in the expression with '-\frac{1}{5}'. Doing this correctly is crucial; even a small mistake can throw off the entire solution. After replacing 'x' with its value, the expression should be simplified, which leads us into our next concept, the simplification of algebraic expressions.
Algebraic Expressions Simplification
Once we have substituted the variables with their respective values, we move on to simplifying the algebraic expression. Simplifying might involve various arithmetic operations: addition, subtraction, multiplication, division, or combination of these. The aim is to break down the expression into its simplest form or a single number if possible.
In our case, after substituting \(x\) with \(-\frac{1}{5}\), we get the expression \(-5(-\frac{1}{5}) - 2\). Multiplying inside the parentheses should be done first, according to the order of operations (PEMDAS/BODMAS). This yields 1 because \(-5\) multiplied by \(-\frac{1}{5}\) is \(1\). Then, we continue by performing the subtraction: 1 - 2 equals -1. Simplifying algebraic expressions requires careful attention to the order of operations to ensure the correct simplification and, consequently, the correct answer.
In our case, after substituting \(x\) with \(-\frac{1}{5}\), we get the expression \(-5(-\frac{1}{5}) - 2\). Multiplying inside the parentheses should be done first, according to the order of operations (PEMDAS/BODMAS). This yields 1 because \(-5\) multiplied by \(-\frac{1}{5}\) is \(1\). Then, we continue by performing the subtraction: 1 - 2 equals -1. Simplifying algebraic expressions requires careful attention to the order of operations to ensure the correct simplification and, consequently, the correct answer.
Other exercises in this chapter
Problem 5
Plot and label the ordered pairs in a coordinate plane. $$ A(-2,-3), B(-3,-2) $$
View solution Problem 6
Determine whether the inequality is a multi-step inequality. Then explain how you would solve the inequality. $$ 4 y-3
View solution Problem 6
The variables \(x\) and \(y\) vary directly. Use the given values to write an equation that relates \(x\) and \(y .\) $$ x=1, y=2 $$
View solution Problem 6
Plot the points and draw the line that passes through them. Without finding the slope, determine whether the slope is positive, negative, zero, or undefined. \(
View solution