Is an article required before a math variable?

This is answered by recognising that L_c and f_t (or f_r) are names, referring to the quantities "channel length" and "transistor frequency" (and whatever the other one is).

You could call "channel length" Cuthbert. A sentence referring to Cuthbert wouldn't use an article:

Fred is the same for both cases at various values of Cuthbert. As Cuthbert increases, Fred also shifts.

Don't use an article for the names/symbols for the quantities.

There's no weirdness in starting a sentence with a name, however it's spelled. If it doesn't have a capital letter, don't use one. It's unusual, because names usually have a capital letter and sentences should start with one. But changing f_t into F_t could substantially change what you are referring to. That is far more important than stylistic custom concerning capital letters. However, many don't like this flouting of convention, in which case, it's necessary to use "The value of f_t ..."

See also the ELU question Is it okay to start a sentence with a Greek letter (variable)?

If f_t, f_r and L_c have been defined previously, the second sentence is clear, unambiguous, conventional and correct.

In mathematical and physical discourse, a function of a variable may be represented as a symbol. For example, we have your f_t , or f(t) as it might often be represented. The value f of the function depends on the value of the determining variable t. The symbol f_t represents the manner in which the value f of the function depends on the determining variable t. Once defined the symbol is understood as the representation of the functional relationship between the two. It is therefore redundant to use "The" with such defined symbols, whether or not they occur at the start of the sentence.

The alphanumeric expression of the function is irrelevant, as is the number of determining variables. This argument applies to any function. For example F(t), f_t, p_b, Φ_x, Gamma(x,y,z),Phi_μ.

It is correct usage to refer to the function f(t) in examples such as the following. In each case the function is being defined on its first use.

"Consider the function f(t)=t*t, where f has a quadratic dependence on t."

"The function sin(t) has a regular periodicity in t"

As with many questions, the answer is consult your style guide.

Here is accepted answer to Is it okay to start a sentence with a Greek letter (variable)?. Note that it is asking the opposite of your question.

• Maybe for Physical Review Style and Notation Guide (only if it didn't end with a symbol or number)
• Both Yes and No for the AMS A manual for authors of mathematical papers
• No, an article is required for Journals of the London Mathematical Society: house style and instructions for copy-editors and typesetters (BrE)

Another answer cites Chicago Manual of Style (but only addresses symbols):

• No, an article is required for Chicago Manual of Style (AmE)

But you are asking about about starting with a (non-Greek) variable name. Both my undergrad professor, Jon Louis Bentley, and my senior editor at IBM, were adamantly opposed to starting with a variable.

In the absence of a clarifying style guide, I would agree with the "weirded out" impression and go with the CMoS, 13th ed., advice for numbers, substituting variables:

At the beginning of a sentence any number that would ordinarily be set in figures is spelled out, regardless of any inconsistency this may create. (Section 8.9)

If this is impracticable or cumbersome, the sentence should be recast so that it does not begin with a number. (Section 8.10)

While it may be acceptable (or your style guide may be mute) to start with a variable, recast the sentence so that it does not start with a variable.