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Chapter 3: Problem 52

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

Short Answer

Expert verified
The value of \( g(3) \) is 8.

Step by step solution

01

Understanding the Problem

The exercise asks us to evaluate the expression \( g(3) \) using the provided function \( g(x) = 5x - 7 \). This means we need to substitute \( x = 3 \) into the function \( g(x) \) and simplify to find the result.
02

Substitute x with 3 in g(x)

Take the function \( g(x) = 5x - 7 \) and replace \( x \) with 3. This gives us the expression \( g(3) = 5(3) - 7 \).
03

Simplify the Expression

Now simplify the expression \( 5(3) - 7 \). First, multiply 5 by 3 to get 15. So, the expression becomes \( 15 - 7 \).
04

Perform the Subtraction

Subtract 7 from 15 to simplify the expression further. This gives us \( 15 - 7 = 8 \). Thus, the value of \( g(3) \) is 8.

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Key Concepts

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

Substitution Method
The substitution method is a key strategy in mathematics used to evaluate functions, solve equations, and simplify expressions. In our exercise, we used substitution to find the value of the function \( g(x) = 5x - 7 \) by replacing the variable \( x \) with the number 3.Here's how it works:
  • Identify which function you need to evaluate, in this case, \( g(x) \).
  • Next, substitute the given value where \( x \) appears in the function. So, \( x \) is replaced with 3.
  • You end up with \( g(3) = 5(3) - 7 \).
By substituting the specific value into the function, you convert the abstract expression into a concrete number. This approach allows you to compute the result and provides a clearer understanding of how input values affect the output.
Linear Functions
Linear functions are a type of function where each point creates a straight line when graphed. They are expressed in the form \( f(x) = mx + c \) or \( g(x) = mx - c \), involving:
  • \( m \) representing the slope: This dictates how steep the line is.
  • \( c \) or \(-c\) representing the y-intercept: This is where the line crosses the y-axis.
In our example with the function \( g(x) = 5x - 7 \):
  • \( m = 5 \), indicating the line rises five units vertically for each unit increased horizontally.
  • \( -7 \) shows the line crosses the y-axis at -7.
Linear functions are straightforward to work with because they have a constant rate of change, which means the same steps can be applied uniformly across the function.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form. This process is crucial for solving functions and equations efficiently.Steps to simplify include:
  • Perform arithmetic operations: In our exercise, we started with \( g(3) = 5\times 3 - 7 \).
  • Multiply \( 5 \times 3 \) to get 15.

  • Subtract: Next step is \( 15 - 7 \), which simplifies to 8.
It is important to execute these arithmetic operations accurately along the order they appear, typically following the order of operations - parentheses, exponents, multiplication, division, and lastly, addition and subtraction. By properly simplifying expressions step by step, we can arrive at correct answers quickly and efficiently.

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Most popular questions from this chapter

Graph each inequality. \(5 y-4 x<-20\)

Create a system of inequalities that forms a bounded region.

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. $$ \begin{array}{l}{y \geq 1} \\ {x \leq 6} \\ {y \leq 2 x+1} \\ {f(x, y)=x+y}\end{array} $$

Solve each system of equations. \(r-3 s+t=4\) \(3 r-6 s+9 t=5\) \(4 r-9 s+10 t=9\)

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. $$ \begin{array}{l}{y \geq-x+3} \\ {1 \leq x \leq 4} \\ {y \leq x+4} \\ {f(x, y)=-x+4 y}\end{array} $$
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