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Chapter 3: Problem 35

Given the following functions, find the function values. \(g(x)=-18 x+58,\) find \(g(58)\).

Short Answer

Expert verified
The value of \(g(58)\) is \(-986\).

Step by step solution

01

Identify the Function

The function provided is \(g(x) = -18x + 58\). This is a linear function, where \(-18\) is the coefficient of \(x\) and \(58\) is the constant term.
02

Substitute the Given Value

Substitute \(x = 58\) into the function. This means wherever you see \(x\) in the equation \(g(x) = -18x + 58\), replace it with \(58\).
03

Perform the Calculation

Calculate \(g(58) = -18(58) + 58\). First, multiply \(-18\) by \(58\) to get \(-1044\). Then, add \(58\) to \(-1044\) to find the result.
04

Simplify the Expression

Simplify \(-1044 + 58\) to get the final value. This results in \(-986\). Thus, \(g(58) = -986\).

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Key Concepts

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

Linear Functions
Linear functions are a foundational concept in mathematics, often expressed in the form \( f(x) = mx + b \). Here:
  • \( m \) is the slope of the line, indicating its steepness.
  • \( b \) is the y-intercept, which is where the line crosses the y-axis.
In the function \( g(x) = -18x + 58 \), \(-18\) represents the slope, showing us that for every unit increase in \( x \), the value of \( g(x) \) decreases by 18 units. The constant \( 58 \) is the y-intercept, indicating that when \( x = 0 \), the value of \( g(x) \) is 58. Linear functions graph as straight lines, making them easy to understand and compute.
Substitution Method
The substitution method is a technique used to evaluate functions by replacing the variable with a specific value. When you're faced with a function like \( g(x) = -18x + 58 \) and need to find \( g(58) \), substitution becomes helpful.Here's how it works:
  • Identify the variable (in this case, \( x \)).
  • Replace every instance of the variable with the specified value (\( x = 58 \)).
For example, substituting \( 58 \) for \( x \) in \( g(x) = -18x + 58 \) changes the function to \( g(58) = -18 \times 58 + 58 \).The substitution method simplifies the function, allowing us to perform straightforward calculations to reach a result.
Mathematical Computation
Mathematical computation refers to the process of performing arithmetic operations to find a result. In the context of evaluating a linear function like \( g(58) = -18 \times 58 + 58 \), it involves:
  • First, doing the multiplication: \(-18 \times 58 = -1044\)
  • Then, performing addition: \(-1044 + 58 = -986\)
These computations follow the order of operations (PEMDAS/BODMAS), ensuring multiplication is done before addition. By calculating \( g(58) \), students practice these operations, reinforcing their understanding of arithmetic within algebra.

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