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Chapter 2: Problem 18

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following $$ g(1.5) $$

Short Answer

Expert verified
g(1.5) = 4.75

Step by step solution

01

Substitute the value into the function

To find the value of the function at a specific point, substitute that point into the given function. Here, substitute 1.5 into the function g(1.5) = - (1.5)^2 + 4(1.5) + 1.
02

Simplify the exponent

Calculate the square of 1.5.g(1.5) = - (1.5)^2 + 4(1.5) + 1g(1.5) = - (2.25) + 4(1.5) + 1.
03

Perform multiplications

Next, multiply the constants to simplify the equation further.g(1.5) = - 2.25 + 6 + 1.
04

Simplify the expression

Combine the terms to find the value of g(1.5).g(1.5) = - 2.25 + 6 + 1 = 4.75

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Key Concepts

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

Substitution in Functions
Understanding substitution in functions is a crucial step in solving algebraic problems. When you have a function, say, like our given function \( g(x) = -x^2 + 4x + 1 \), you can find the value of the function at a specific point by substituting the given value into the function. In this exercise, we need to find \( g(1.5) \). This means replacing every occurrence of \( x \) in the function with 1.5.
So, the substitution looks like this: `g(1.5) = - (1.5)^2 + 4(1.5) + 1`.
This first step is essential because it sets up the entire problem for simplification and solution, allowing us to find the function's value at that particular point.
Simplifying Exponents
After substituting the value, the next step involves simplifying any exponents in the equation. Specifically, we need to calculate \( (1.5)^2 \).
An exponent tells us how many times to use the number in a multiplication. So, \( (1.5)^2 \) means multiplying 1.5 by itself: \( 1.5 \times 1.5 = 2.25 \).
This can be plugged back into the function for further simplification: `g(1.5) = - 2.25 + 4(1.5) + 1`.
Simplifying exponents early in the process helps to simplify calculations later, making the overall problem more manageable.
Performing Multiplications
Once the exponents are simplified, the next task is to perform any remaining multiplications. In our function, we need to multiply the coefficients by the substituted value:
`4(1.5)`.
To do this, multiply 4 by 1.5: 4 times 1.5 equals 6.
Now, our equation looks like this: `g(1.5) = -2.25 + 6 + 1`.
Performing these multiplications makes the combining of like terms simpler and the expression easier to manage.
Combining Like Terms
The final step is to combine like terms to produce the simplified result. Like terms are terms in an expression that have the same variable raised to the same power, or are simply constants. In our case, we add or subtract the constants: `-2.25 + 6 + 1`.
This involves straightforward addition and subtraction:
  • First, add 6 and 1 to get 7
  • Next, subtract 2.25 from 7, resulting in 4.75

So, `g(1.5) = 4.75`.
Combining like terms is the final crucial step in simplifying the function and finding the value at the given point.

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Most popular questions from this chapter

A factory can have no more than 200 workers on a shift, but must have at least 100 and must manufacture at least 3000 units at minimum cost. How many workers should be on a shift in order to produce the required units at minimal cost? Let \(x\) represent the number of workers and y represent the number of units manufactured. Write three inequalities expressing the problem conditions.

Graph each line passing through the given point and having the given slope. (-3,1)\(;\) undefined slope

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

Find the slope of each line in three ways by doing the following. (a) Give any two points that lie on the line, and use them to determine the slope. (b) Solve the equation for \(y\), and identify the slope from the equation. (c) For the form \(A x+B y=C,\) calculate \(-\frac{A}{B} .\) x+y=-3

An equation that defines \(y\) as a function \(f\) of \(x\) is given. (a) Solve for \(y\) in terms of \(x\), and write each equation using function notation \(f(x) .\) (b) Find \(f(3)\). $$ y-3 x^{2}=2 $$
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