Can a single variable function equal a multivariable one? In the book 'Calculus, Early Transcendentals' by James Stewart, the author writes at some point:
$$ g(x)=f(x,b) $$
when taking the partial of $f$ with respect to $x$. Now, this is a reasonable thing to assume (In fact, when I take partial derivatives I think exactly like that). That made me ask the question: can multi-variable function ever equal a single variable one ? $f(x,b)$ is  a function that takes as input order tuples $(x,b)$ (with $b$ being a constant) and outputs a real value. On the other hand, $g(x)$ takes only a real value and outputs a real value. The domains of the two functions could look for example like:
$$ D_{f} = \{(x,y)\in \mathbb R \times\{ b\}  | \ x\in\mathbb R \} , \ b\in\mathbb R $$
and
$$ D_g = \mathbb R$$
Obviously, $D_{f} \neq D_g $ (The former contains tuples while the latter does not) which means that they can't be equal functions.
 A: The other answers are misleading. The answer to your question:

Can a single variable function equal a multivariable one?

is no taken literally. You cannot set a multivariable function to be equal to a single variable function --- but this is not what the equation is saying. The equation
$$
g(x) = f(x, b)
$$
should be understood as saying: for all values of $x$, the value of $g(x)$ is equal to the value $f(x, b)$.
So it is not a statement about equality between two functions, but equality of real numbers. And of course it is totally fine to write a statement of equality between real numbers.
More generally, when we write equations between applications of functions of one or more variables ($x$ and $y$ etc.), we generally mean this should hold for all values of the variables. So for instance, $\sin (-x) = \sin x$ is an equation of two functions of $x$, which should hold for all values of $x$.
A: The answer to your question is yes, and Pink Panther's comment makes it clear why.
You might find it instructive to observe that good Computer Algebra Systems allow for this, e.g. http://magma.maths.usyd.edu.au/magma/handbook/text/226#1986
'IsUnivariate(f) : RngMPolElt -> BoolElt, RngUPolElt, RngIntElt
Given a multivariate polynomial f∈R[x1, ..., xn], this function returns whether f is in fact a univariate polynomial in one of its indeterminates x1, ..., xn. If true is returned, then the function also returns a univariate version u of f and (the first) i such that f is univariate in xi. Note that there will only be ambiguity about i if f is a constant polynomial. The univariate polynomial u will be an element of R[x] with the same coefficients as f.'
