Problem 10 Let $f(x)=-3 x+4$ and \(g(x)=-... [FREE SOLUTION]

91影视

Chapter 2: Problem 10

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

Short Answer

Expert verified

The value of $g(-1)$ is $-4$.

Step by step solution

- Identify the given function

The function we need to evaluate is given as $g(x) = -x^2 + 4x + 1$.

- Substitute the value into the function

Substitute $x = -1$ into the function $g(x)$. This gives us: \[ g(-1) = -(-1)^2 + 4(-1) + 1 \]

- Simplify the expression

First, calculate $(-1)^2$: \[ g(-1) = -1 + 4(-1) + 1 \] Next, calculate $4(-1)$: \[ g(-1) = -1 - 4 + 1 \] Finally, add the numbers together: \[ g(-1) = -4 \- 4 + 1 = -4 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

substitution in functions

Substituting values into functions is a common task in mathematics. It entails replacing the variable with a given number and then simplifying the resulting expression.

To substitute a value into a function, follow these easy steps:

Identify the function and the value to substitute.
Replace the variable in the function with the given value.
Simplify the resulting expression carefully.

In our example, we have the quadratic function $ g(x) = -x^2 + 4x + 1 $. By substituting $ x = -1 $, we get:
\[ g(-1) = -(-1)^2 + 4(-1) + 1 \]
This step sets the stage for further simplification.

simplifying expressions

Simplifying expressions is key to solving mathematical problems. Here are the steps for simplification in our example:

Calculate the exponent, if any.
Perform any multiplication or division.
Combine like terms by addition or subtraction.

When we substitute $ x = -1 $ into our function, we start simplifying:
\[ g(-1) = -(-1)^2 + 4(-1) + 1 \]
First, calculate $(-1)^2$, which equals 1:
\[ g(-1) = -1 + 4(-1) + 1 \]
Next, multiply 4 and -1, getting -4:
\[ g(-1) = -1 - 4 + 1 \]
Finally, add the numbers together step-by-step:
\[ -1 - 4 + 1 = -4 \]
Thus, simplifying helps us find the final value, $ g(-1) = -4 $.

quadratic functions

Quadratic functions are polynomials of degree 2 and have the general form $ ax^2 + bx + c $. They are important in many areas of mathematics and are defined by three coefficients:

$ a $: the coefficient of $ x^2 $
$ b $: the coefficient of $ x $
$ c $: the constant term

The function $ g(x) = -x^2 + 4x + 1 $ is a quadratic function.

Quadratic functions typically have a parabolic shape when graphed: if $ a > 0 $, the parabola opens upwards; if $ a < 0 $, it opens downwards.

To evaluate a quadratic function at a specific point, substitute the value of $ x $ and simplify. This process, seen in our example, helps find specific values and understand the function's behavior at different points.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

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

Short Answer

Step by step solution

- Identify the given function

- Substitute the value into the function

- Simplify the expression

Key Concepts

substitution in functions

simplifying expressions

quadratic functions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Decision Maths

Statistics

Mechanics Maths

Geometry

Discrete Mathematics

Study anywhere. Anytime. Across all devices.