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Chapter 3: Problem 44

Find each value if \(f(x)=6 x+2\) and \(g(x)=3 x^{2}-x\). \(g(1)\)

Short Answer

Expert verified
The value of \( g(1) \) is 2.

Step by step solution

01

Identify the Function

The function given for this exercise is \( g(x) = 3x^2 - x \). We need this function to find \( g(1) \).
02

Substitute the Value into the Function

Substitute the value \( x = 1 \) into the function \( g(x) = 3x^2 - x \). So, calculate \( g(1) = 3(1)^2 - 1 \).
03

Calculate the Expression

First, compute \( 1^2 \), which is \( 1 \). Then multiply by 3 to get \( 3 \times 1 = 3 \).
04

Simplify the Function Evaluation

Subtract the value of \( x = 1 \) from \( 3 \). Therefore, \( 3 - 1 = 2 \).

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Key Concepts

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

Substitution in Algebra
Substitution in algebra involves replacing a variable in an expression with a specific value to simplify the expression or solve an equation. This process is fundamental in solving function evaluations. For instance, if we have a function \( g(x) = 3x^2 - x \) and we need to find \( g(1) \), we're using substitution.
The steps involved are simple:
  • Identify the variable in the expression. In our function, \( x \) is the variable.
  • Replace the variable with the given number. Here, we replace \( x \) with 1.
  • Recalculate using the substituted value to find the result.
By substituting \( x = 1 \) into \( g(x) \), we perform calculations step-by-step, ensuring accuracy, and finally arrive at a numerical answer. This method allows us to solve for specific outputs where the input is known.
Polynomial Functions
Polynomial functions are algebraic expressions involving a sum of powers in one or more variables multiplied by coefficients. In simpler terms, these are expressions made up of terms like \( ax^n \), where \( a \) is a coefficient, \( x \) is the variable, and \( n \) is a non-negative integer called the degree of the term.
Our function \( g(x) = 3x^2 - x \) is a polynomial. It consists of:
  • A quadratic term \( 3x^2 \), where the degree of the term is 2 because of the exponent on \( x \).
  • A linear term \( -x \), as it has a degree of 1.
Polynomial functions are versatile and fundamental in algebra, allowing us to model a wide range of scenarios and behaviors. They help in understanding trends in data and predicting future outcomes by analyzing the relation between variables.
Quadratic Expressions
Quadratic expressions are a specific type of polynomial where the highest exponent of the variable is 2. These expressions are typically of the form \( ax^2 + bx + c \), with \( a \), \( b \), and \( c \) as constants and \( a eq 0 \).
In the function \( g(x) = 3x^2 - x \), the part \( 3x^2 \) represents the quadratic term. Quadratic expressions usually form a U-shaped curve called a parabola when graphed. Some characteristics of quadratics are:
  • The parabola can open upward or downward, depending on the sign of \( a \). Here, since \( a = 3 \), it opens upwards.
  • The vertex, which is the turning point of the parabola, provides critical information about the maximum or minimum value of the function.
Quadratics are used extensively in solving real-world problems involving area calculations, projectile motions, and economics.

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Most popular questions from this chapter

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. $$ \begin{array}{l}{y \geq 1} \\ {2 \leq x \leq 4} \\ {x-2 y \geq-4} \\ {f(x, y)=3 y+x}\end{array} $$

Find each value if \(f(x)=6 x+2\) and \(g(x)=3 x^{2}-x\). \(g(-3)\)

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. $$ \begin{array}{l}{2 x+3 y \geq 6} \\ {3 x-2 y \geq-4} \\ {5 x+y \geq 15} \\\ {f(x, y)=x+3 y}\end{array} $$

FARMING For Exercises \(34-37\) , use the following information. Dean Stadler has 20 days in which to plant corn and soybeans. The corn can be planted at a rate of 250 acres per day and the soybeans at a rate of 200 acres per day. He has 4500 acres available for planting these two crops. If the profit is \(\$ 26\) per acre on corn and \(\$ 30\) per acre on soybeans, how much of each should Mr. Stadler plant? What is the maximum profit?

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