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Chapter 3: Problem 88

Given the following, \(f(x)=-5 x+1,\) find \(f(-3)\)

Short Answer

Expert verified
The value of \(f(-3)\) is 16.

Step by step solution

01

Identify the Function

We are given the function, \(f(x) = -5x + 1\). This is a linear function with a slope of \(-5\) and a y-intercept of \(1\).
02

Substitute the Value into the Function

To find \(f(-3)\), substitute \(-3\) for \(x\) in the function: \(f(-3) = -5(-3) + 1\).
03

Simplify the Expression

Simplify the expression obtained from substitution: \(-5(-3) = 15\). So, \(f(-3) = 15 + 1\).
04

Perform the Final Calculation

Add 1 to 15: \(f(-3) = 15 + 1 = 16\). Therefore, the value of the function at \(x = -3\) is 16.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Evaluation
When working with functions in mathematics, function evaluation is a key concept. It involves determining the output or value of a function at a given input value. In our problem, we're given the function \(f(x) = -5x + 1\), and we're asked to find \(f(-3)\). This means we want to evaluate the function at \(x = -3\). Function evaluation helps us understand how the function behaves for specific inputs, which is crucial in various applications like economics, physics, and engineering. Evaluating a linear function like this one involves straightforward calculations, making it an ideal starting point for learning the basics of functions.
Substitution Method
The substitution method is a straightforward way to evaluate functions, especially linear ones. To use this method, you replace the variable \(x\) in the expression \(-5x + 1\) with \(-3\). This gives us \(f(-3) = -5(-3) + 1\).
  • Find the replacement value, in this case, \(-3\).
  • Replace all instances of \(x\) with \(-3\) in the expression.
By substituting \(-3\) into the function equation, you align the problem to a stage where simple arithmetic can solve it. The substitution method turns the function problem into a more familiar form of mathematics鈥攑ure arithmetic!
Simplification Process
Simplification is the next essential step after substitution. It involves performing arithmetic operations to tidy up the expression. Once you have the substituted expression \(-5(-3) + 1\), you perform operations:
  • Multiply: \(-5 \times -3 = 15\).
  • Addition: \(15 + 1 = 16\).
This step ensures that the expression is reduced to its simplest form, making it easy to understand the function's value at the specified input. Simplification affects the comprehension of functions by stripping away extra complexity and focusing on the numerical value.
Slope and Y-intercept
In linear functions, the slope and y-intercept are fundamental features that describe the line on a graph. For \(f(x) = -5x + 1\), the slope is \(-5\), indicating a downward trend鈥攖he function decreases five units vertically for every one unit increase horizontally. The y-intercept is \(1\), where the line crosses the y-axis. This function's behavior can be depicted visually as a line sloping downwards, starting from the point \(1\) on the y-axis. Understanding these components helps predict and interpret the function's graph even without calculating every point.

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