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Chapter 3: Problem 53

Find each value if \(f(x)=4 x+3\) and \(g(x)=5 x-7\). $$ g(-0.25) $$

Short Answer

Expert verified
The value is \(-8.25\).

Step by step solution

01

Understand the Function

Recognize that you need to find the value of \(g(x)\) when \(x = -0.25\). The function given is \(g(x) = 5x - 7\).
02

Substitute the Value

Substitute \(-0.25\) into the function \(g(x)\). This gives us: \[ g(-0.25) = 5(-0.25) - 7 \]
03

Calculate the Expression inside the Function

First, calculate \(5 \times (-0.25)\). Multiplying gives: \[ 5 \times (-0.25) = -1.25 \]
04

Simplify the Result

Now replace \(5 \times (-0.25)\) with \(-1.25\) in the equation: \[ g(-0.25) = -1.25 - 7 \] Then do the subtraction: \[ -1.25 - 7 = -8.25 \]
05

Conclusion

The value of \(g(-0.25)\) is \(-8.25\).

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Key Concepts

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

Substitution
Substitution is a crucial technique in mathematics used to evaluate functions. It's like replacing a placeholder with a specific value to simplify and solve expressions. In the context of our exercise, we were tasked with finding the value of the function \( g(x) = 5x - 7 \) when \( x = -0.25 \). This means we substitute \( x \) with \( -0.25 \) in the expression.

To perform substitution, follow these steps:
  • Identify the variable in the function that needs replacing, which in our case is \( x \).
  • Carefully replace each instance of this variable in the function with the given number \( -0.25 \).
  • Once replaced, solve the resulting expression step by step to find the desired value.
Substitution might seem simple, but it's the foundation of evaluating and manipulating mathematical expressions.
Linear Functions
Linear functions are a fundamental concept in mathematics, often introduced in early algebra courses. They have the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants.

In our exercise, the function \( g(x) = 5x - 7 \) is a perfect example of a linear function. **Key characteristics of linear functions include:**
  • They graph as straight lines on a coordinate plane.
  • The coefficient \( a \) (in this case, 5) represents the slope or steepness of the line.
  • The constant \( b \) (here, -7) indicates the Y-intercept, where the line crosses the Y-axis.
Understanding linear functions is important as they represent many real-world relationships where quantities change at constant rates.
Function Notation
Function notation is a way to express the relationship between input values (often \( x \)) and output values (\( f(x) \) or \( g(x) \), etc.). It allows us to represent functions in a concise and formal manner.

**Points to note about function notation:**
  • The letter outside the parentheses (for example, \( g \) in \( g(x) \)) is often a way of naming the function.
  • The variable inside the parentheses (\( x \) in this case) represents the input or independent variable.
  • This notation provides a clear and organized way to substitute different vales into the function, allowing for the calculation of various outputs.
In our exercise, \( g(-0.25) = 5(-0.25) - 7 \) uses function notation to indicate that we are finding the output when \( x = -0.25 \). This notation is beneficial for keeping calculations organized and readable, especially in more complex mathematical problems.

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Most popular questions from this chapter

Evaluate each expression if \(x=-2, y=6,\) and \(z=5\) $$ 3 x-y+4 z $$

Solve each system of inequalities by graphing. \(3 x+y \geq 1\) \(2 y-x \leq-4\)

Graph each system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the given function for this region. $$ \begin{array}{l}{y \geq-x+3} \\ {1 \leq x \leq 4} \\ {y \leq x+4} \\ {f(x, y)=-x+4 y}\end{array} $$

In order to practice at home, Tadeo purchased a basketball and a volleyball that cost a total of \(\$ 67,\) not including tax. If the price of the basketball \(b\) is \(\$ 4\) more than twice the cost of the volleyball \(v\) which system of linear equations could be used to determine the price of each ball? A. \(b+v=67\) \(b=2 v-4\) B. \(b+v=67\) \(b=2 v+4\) C. \(b+v=4\) \(b=2 v-67\) D. \(b+v=4\) \(b=2 v+67\)

Graph each system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the given function for this region. $$ \begin{array}{l}{2 x+2 y \geq 4} \\ {2 y \geq 3 x-6} \\ {4 y \leq x+8} \\\ {f(x, y)=3 y+x}\end{array} $$
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