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Chapter 2: Problem 17

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following $$ g(0.5) $$

Short Answer

Expert verified
2.75

Step by step solution

01

Identify the Function to Evaluate

Given the function to evaluate is \( g(x) = -x^2 + 4x + 1 \).
02

Substitute the Given Value

Substitute \( x = 0.5 \) into the function \( g(x) \).
03

Simplify the Expression

Calculate \( g(0.5) \) by substituting and simplifying: \( g(0.5) = -(0.5)^2 + 4(0.5) + 1 \).
04

Perform the Calculations

Solve the equation step by step: 1. \( (0.5)^2 = 0.25 \) 2. \(-0.25 + 2 + 1 \) 3. \(2.75 \).

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Key Concepts

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

substitute value
To solve a function such as the one in the given exercise, substituting the correct value is the first vital step.
Substituting a value means to replace the variable (usually x) in the function with a specific number.
For example, in the function \( g(x) = -x^2 + 4x + 1 \), we are asked to find \( g(0.5) \).
This means replacing every instance of 'x' in the function with 0.5.
simplify expression
Once you have substituted the value into the function, the next step is to simplify the expression.
Simplifying means performing all necessary mathematical operations to reduce the expression to a single value.
In this case, you'd first square 0.5, then multiply accordingly, and finally, sum and subtract the results.
For example, substituting in & simplifying \( g(0.5) = -(0.5)^2 + 4(0.5) + 1 \) would involve:
  • Calculating \( (0.5)^2 = 0.25 \).
  • Applying the negative sign to get \( -(0.25) = -0.25 \).
  • Calculating \( 4(0.5) = 2 \).
  • Adding the results: \( -0.25 + 2 + 1 = 2.75 \).
This breakdown makes the expression easier to handle step-by-step.
quadratic function
A quadratic function is a type of polynomial that can be written in the form \( ax^2 + bx + c \), where a, b, and c are constants, and x represents an unknown variable.
The 'u'-shaped graph of a quadratic function is called a parabola.
The function \( g(x) = -x^2 + 4x + 1 \) is quadratic because it fits this form with \( a = -1 \), \( b = 4 \), and \( c = 1 \).
Quadratic functions are valuable because they model many real-world situations and can show maximum or minimum points, important in optimization problems.
Understanding how to substitute values and simplify expressions is crucial for solving quadratic functions like the one in the exercise.

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Most popular questions from this chapter

Determine whether each pair of lines is parallel, perpendicular, or neither. \(4 x-3 y=6\) and \(3 x-4 y=2\)

For each function, find (a) \(f(2)\) and (b) \(f(-1)\) $$ \begin{array}{c|c} x & y=f(x) \\ \hline 2 & 4 \\ \hline 1 & 1 \\ \hline 0 & 0 \\ \hline-1 & 1 \\ \hline-2 & 4 \end{array} $$

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