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Chapter 9: Problem 12

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1\) \(g(10)\)

Short Answer

Expert verified
\( g(10) = -59 \)

Step by step solution

01

Identify the given function

The exercise provides the function: \[ g(x) = -x^2 + 4x + 1 \]
02

Substitute the value into the function

To find \( g(10) \), substitute \( x = 10 \) into the function: \[ g(10) = -10^2 + 4(10) + 1 \]
03

Calculate the squared term

Compute \( 10^2 \): \[ 10^2 = 100 \]Then: \[ -10^2 = -100 \]
04

Calculate the multiplication term

Next, compute \( 4 \times 10 \): \[ 4 \times 10 = 40 \]
05

Combine all terms

Now, combine the calculated terms together: \[ g(10) = -100 + 40 + 1 \]
06

Perform the final addition

Add the terms together to get the final result: \[ g(10) = -100 + 40 + 1 = -59 \]

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Key Concepts

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

Substitution
Substitution is a fundamental technique in algebra that involves replacing a variable with a given number. In our exercise, we needed to find the value of the function \( g(x) \) at \( x = 10 \). To do this, we substituted 10 for the variable \( x \) in the function \( g(x) = -x^2 + 4x + 1 \). This gave us the new expression \( g(10) = -10^2 + 4(10) + 1 \). Using substitution allows us to transform the function into an equation that we can solve step-by-step.
Quadratic Functions
Quadratic functions are polynomial functions of degree 2 and have the general form \( ax^2 + bx + c \). In our case, the function \( g(x) = -x^2 + 4x + 1 \) is a quadratic function where the coefficient \( a = -1 \), \( b = 4 \), and \( c = 1 \). Quadratic functions often exhibit a parabolic shape on a graph, opening upwards if \( a > 0 \) and downwards if \( a < 0 \). When evaluating a quadratic function at a specific point, we substitute the value into the function and perform the necessary calculations, just as we did when we found \( g(10) \).
Step-by-Step Calculation
A step-by-step calculation helps break down complex problems into manageable parts. Let's revisit our example:

  • First, identify the given function: \( g(x) = -x^2 + 4x + 1 \)
  • Next, substitute 10 for \( x \): \( g(10) = -10^2 + 4(10) + 1 \)
  • Then, calculate the squared term: \( 10^2 = 100 \) followed by \( -10^2 = -100 \)
  • After that, compute the multiplication term: \( 4 \times 10 = 40 \)
  • Now, combine all the calculated terms: \( g(10) = -100 + 40 + 1 \)
  • Finally, perform the addition to find the result: \( g(10) = -100 + 40 + 1 = -59 \)
This structured approach ensures you don鈥檛 miss any steps and helps maintain clarity and precision throughout the problem-solving process.

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Most popular questions from this chapter

Let \(f(x)=x^{2}\) and \(g(x)=2 x-1 .\) Match each expression in Column \(I\) with the description of how to evaluate it in Column II. II A. Square \(5 .\) Take the result and square it. B. Double 5 and subtract \(1 .\) Take the result and square it. C. Double 5 and subtract 1 . Take the result, double it, and subtract 1 . D. Square \(5 .\) Take the result, double it, and subtract 1 . I $$ (g \circ g)(5) $$

Let \(f(x)=x^{2}+4, g(x)=2 x+3,\) and \(h(x)=x-5 .\) Find each of the following. $$ (f \circ h)(0) $$

For each pair of functions, find \(\left(\frac{f}{g}\right)(x)\) and give any \(x\) -values that are not in the domain of the quotient function. $$ f(x)=27 x^{3}+64, \quad g(x)=3 x+4 $$

Determine whether each relation defines y as a function of \(x .\) (Solve for y first if necessary.) Give the domain. $$ x y=1 $$

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ \left(\frac{h}{g}\right)(3) $$
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