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Chapter 1: Problem 48

Evaluate the function for the given value of \(x\). \(f(x)=-x+3 ; x=5\)

Short Answer

Expert verified
The function \(f(5)\) evaluates to -2.

Step by step solution

01

Substitute the given value into the function

Replace \(x\) in \(f(x)=-x+3\) with the given value \(x=5\). It then becomes \(f(5)=-5+3\).
02

Simplify the equation

After substituting, the equation simplifies to \(f(5)=-2\).

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Key Concepts

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

Substitution Method
When evaluating functions, the substitution method is a fundamental tool. It involves replacing the variable in a function with a given number or expression. This technique is crucial in mathematics as it enables us to find out what the function's output is for any specific input.

  • First, identify the variable in the function. In the example, the variable is x.
  • Next, replace the variable with the provided number. Here, we substitute x with 5.
  • Apply the substitution across the entire function. In the exercise, substituting the value leads to the function changing from f(x) = -x + 3 to f(5) = -5 + 3.

This process of substitution is at the heart of function evaluation and is used in various fields, from simple algebra to complex calculus.
Function Evaluation
Function evaluation is the process whereby we calculate the output of a function based on the input value. It's like following a recipe to transform ingredients (input) into a dish (output).

  • Start by understanding the function's rule; in this case, it is f(x) = -x + 3.
  • Insert the input value using the substitution method aforementioned.
  • Finally, compute the resulting expression to get your output, which is the function鈥檚 value at the input.

In our example, the output, or the function's value at x = 5, is calculated as f(5) = -2. Recognizing and applying function evaluation is crucial for understanding more advanced mathematics.
Simplifying Expressions
Simplifying expressions is the process of reducing an expression to its simplest form. This makes it easier to understand and work with the expression.

  • To simplify an expression, combine like terms and perform any operations like addition or multiplication.
  • Keep in mind the order of operations: parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right.
  • Simplification can also involve factoring expressions or reducing fractions.

In the example, simplifying the expression after substitution yielded f(5) = -5 + 3 which simplifies to f(5) = -2. Becoming adept at simplifying expressions is key to effectively solving algebraic equations and inequalities.

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Most popular questions from this chapter

Solve the system using the elimination method. \(x+3 y-z=2\) \(x+y-z=0\) \(3 x+2 y-3 z=-1\)

Write a function \(g\) whose graph represents the indicated transformation of the graph of \(f\). Use a graphing calculator to check your answer. \(f(x)=|6 x|-2\); reflection in the \(y\)-axis

Write a function \(g\) whose graph represents the indicated transformation of the graph of \(f\). Use a graphing calculator to check your answer. \(f(x)=|x+3| ;\) horizontal stretch by a factor of 4

Write a function \(g\) whose graph represents the indicated transformation of the graph of \(f\). Use a graphing calculator to check your answer. \(f(x)=-5 x+2\); reflection in the \(x\)-axis

ABSTRACT REASONING The functions \(f(x)=m x+b\) and \(g(x)=m x+c\) represent two parallel lines. a. Write an expression for the vertical translation of the graph of \(f\) to the graph of \(g\). b. Use the definition of slope to write an expression for the horizontal translation of the graph of \(f\) to the graph of \(g\).
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