Question

What will be the domain for which the functions $f(x)=2x^2+1$ and $g(x)=1+3x$ are equal?

• A. $\mathbb{R}$
• B. $\mathbb{R}$
• C. $\mathbb{R}$
• D. $\mathbb{R}$

Answer 1

Answer: (A), (B), (C), (D)

$\mathbb{R}$

Answer 2

The question asks for the domain for which the functions $f(x)=2x^2+1$ and $g(x)=1+3x$ are equal.

There are two common interpretations for this type of question:

1. Interpretation 1: Find the set of all $x$ values for which $f(x) = g(x)$.

   To find these values, we set the expressions for $f(x)$ and $g(x)$ equal to each other:

   $2x^2 + 1 = 1 + 3x$

   Subtract 1 from both sides:

   $2x^2 = 3x$

   Rearrange the terms to form a quadratic equation:

   $2x^2 - 3x = 0$

   Factor out $x$:

   $x(2x - 3) = 0$

   This equation holds true if either $x=0$ or $2x-3=0$.

   Case 1: $x = 0$

   Case 2: $2x - 3 = 0 \implies 2x = 3 \implies x = \frac{3}{2}$

   So, the functions are equal only at $x=0$ and $x=\frac{3}{2}$. If this interpretation is intended, the domain would be $\{0, \frac{3}{2}\}$.

2. Interpretation 2: Find the domain of definition for which both functions are defined.

   This interpretation assumes the question is asking for the common domain over which the functions exist, regardless of whether their values are equal at every point in that domain.

   The function $f(x) = 2x^2+1$ is a polynomial function. Polynomial functions are defined for all real numbers. Therefore, the domain of $f(x)$ is $\mathbb{R}$ (the set of all real numbers).

   The function $g(x) = 1+3x$ is also a polynomial function. Its domain is also $\mathbb{R}$.

   Since both functions are defined for all real numbers, their common domain of definition is $\mathbb{R} \cap \mathbb{R} = \mathbb{R}$.

Given that all the provided options are $\mathbb{R}$, it is highly probable that the question intends to ask for the domain of definition of the functions. While the phrasing "domain for which the functions are equal" can be ambiguous, in the context of multiple-choice questions where only $\mathbb{R}$ is offered as an option for polynomial functions, this interpretation is typically implied. If the question intended the first interpretation, the options would likely include $\{0, \frac{3}{2}\}$.

Therefore, assuming the question asks for the domain of definition of these functions, the answer is $\mathbb{R}$.