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Chapter 7: Problem 4

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following. $$ f(-3) $$

Short Answer

Expert verified
f(-3) = 13

Step by step solution

01

- Understand the function

The function given is f(x) = -3x + 4. This is a linear function where the slope is -3, and the y-intercept is 4.
02

- Substitute the value

Substitute the given value x = -3 into the function f(x). This means replacing x with -3 in the equation.
03

- Perform the calculation

Substitute and calculate: f(-3) = -3(-3) + 4.
04

- Simplify the expression

Simplify the result of the expression calculated in Step 3: f(-3) = 9 + 4 = 13.

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Key Concepts

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

Linear Function
A linear function is a mathematical equation of the form \(f(x) = mx + b\), where:
  • \(m\) is the slope of the line, indicating how steep the line is.
  • \(b\) is the y-intercept, where the line crosses the y-axis.
For example, in the given function \(f(x) = -3x + 4\), \(-3\) is the slope and \(4\) is the y-intercept.
Understanding linear functions is crucial because they graph as straight lines and are foundational to higher-level math concepts.
In real-world contexts, linear functions can represent relationships with constant rates of change, like speed or cost.
Substitution
Substitution is a fundamental step in evaluating functions. It involves replacing a variable with a given value.
In our exercise, we need to find \(f(-3)\). This means we will replace \(x\) with \(-3\) in the function \(f(x) = -3x + 4\).
Substitution helps us find the specific output or value of a function for a particular input. It's like following a recipe where each ingredient (variable) is given a specific amount (value). This step sets the stage for the calculations needed to find the result.
Calculation Steps
After substitution, the next step is performing the actual calculation. Let鈥檚 break it down:
  • Start with the function after substitution: \(f(-3) = -3(-3) + 4\).
  • Multiply: \(-3 \cdot -3 = 9\).
  • Then, add the constant term from the function: \(9 + 4\). This results in \(13\).
Calculations should be done carefully to avoid errors. Each arithmetic operation must follow correct order: multiplication before addition. This ensures the accuracy of the final result.
Simplification
Simplification involves making an expression easier to understand by combining like terms or performing arithmetic operations.
In our example, the expression after substitution and initial calculation is: \(-3(-3) + 4\).
We simplify by:
  • Multiplying to get \(9\).
  • Adding \(4\) to \(9\) to get the final result of \(13\).
Simplification is essential because it reduces complex expressions into simpler forms that are easier to interpret and solve. It brings clarity and accuracy to mathematical problems. We're ensuring we've adhered to the proper mathematical rules and principles during the process.

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Most popular questions from this chapter

Solve each problem. If \(m\) varies inversely as \(p^{2},\) and \(m=20\) when \(p=2,\) find \(m\) when \(p=5\)

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following. $$ f(x+h) $$

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