Problem 22 Let $f(x)=-3 x+4$ and \(g(x)=-... [FREE SOLUTION]

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Chapter 2: Problem 22

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

Short Answer

Expert verified

g(-x) = -x^2 - 4x + 1

Step by step solution

Understand the Function g(x)

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

Substitute -x for x in g(x)

To find $g(-x)$, replace every instance of $x$ with $-x$ in the function $g(x)$. This will give us: $g(-x) = -(-x)^2 + 4(-x) + 1$.

Simplify the Expression

Simplify the expression $-(-x)^2 + 4(-x) + 1$.$-(-x)^2$ simplifies to $-x^2$ because $(-x)^2 = x^2$. Then, $4(-x)$ simplifies to $-4x$. Thus, the expression becomes $-x^2 - 4x + 1$.

Write the Final Answer

The final expression for $g(-x)$ is $-x^2 - 4x + 1$.

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Key Concepts

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

Function Transformation

Function transformation involves modifying the basic form of a function to shift, stretch, compress, or reflect it.
Understanding how transformations impact the graph or equation of a function is essential.
In the given problem, transforming the function involves substituting $x$ with $-x$.
This is a reflection across the y-axis.
By substituting $x$ with $-x$, we create a new function that mirrors the original across the y-axis.
The transformed function can then be simplified accordingly.

Substitution in Functions

Substitution in functions means replacing a variable with a specified value or expression.
In this exercise, we replaced $x$ with $-x$ in the function $g(x) = -x^2 + 4x + 1$.
The steps to substitute $x$ involve identifying every instance of $x$ in the function and replacing it consistently:
\begin{itemize}

Start with $g(x) = -x^2 + 4x + 1$

Replace $-x$: $g(-x) = -(-x)^2 + 4(-x) + 1$

Simplifying Expressions

Simplifying expressions means reducing them to a simpler or more compact form.
It's achieved by performing algebraic operations.
In the given exercise, simplification involves:

Calculating $-(-x)^2$
Simplifying $-(-x)^2$ to $-x^2$ because $(-x)^2 = x^2$
Simplifying \4(-x)\ to $-4x$
Combining all simplified terms to get $-x^2 - 4x + 1$

This step ensures that the substituted function remains in its simplest and most understandable form.

Algebraic Functions

An algebraic function is any function defined by a polynomial expression.
In this case, both $f(x)$ and $g(x)$ are algebraic functions of varying degrees.
The function $g(x) = -x^2 + 4x + 1$ is a quadratic function since it involves $x^2$ (a polynomial of degree 2).
Understanding algebraic functions includes knowing the general form, degree, and how to apply operations to them.
For quadratic functions, transformations like reflections or translations can change their graphs significantly.
These operations can be done by manipulating the algebraic expressions directly.
The resulting expressions still represent the same function type but exhibit different properties.

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91影视

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

Short Answer

Step by step solution

Understand the Function g(x)

Substitute -x for x in g(x)

Simplify the Expression

Write the Final Answer

Key Concepts

Function Transformation

Substitution in Functions

Simplifying Expressions

Algebraic Functions

One App. One Place for Learning.

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