Question Video: Finding an Unknown in a Rational Function given Its Parity Mathematics

Video Transcript

Find the value of 𝑎 given 𝑓 is an even function, where 𝑓 of 𝑥 equals six over eight 𝑥 squared plus 𝑎𝑥 minus three and 𝑥 is not equal to zero.

We’ll begin by thinking about what we know about even functions. Let’s imagine we have two functions 𝑓 sub one and 𝑓 sub two that are even and 𝑓 sub two is not zero. Then the quotient 𝑓 sub one over 𝑓 sub two must also be even. And secondly, we know that a function is said to be even if 𝑓 of negative 𝑥 equals 𝑓 of 𝑥 for all values of 𝑥 in the function’s domain. Let’s define 𝑓 sub one to be the numerator of our expression. That’s six. Now, in fact, this expression is always equal to six no matter the value of 𝑥; it’s entirely independent of 𝑥. And so we can say that 𝑓 sub one is even.

So, for our function 𝑓 of 𝑥 to be even, we need the denominator of our function 𝑓 of two to also be even. So we define 𝑓 of two to be equal to eight 𝑥 squared plus 𝑎𝑥 minus three. And we know that if this is even, 𝑓 sub two of negative 𝑥 must be equal to 𝑓 sub two of 𝑥. Well, 𝑓 sub two of negative 𝑥 is the expression eight times negative 𝑥 squared plus 𝑎 times negative 𝑥 minus three. That simplifies to eight 𝑥 squared minus 𝑎𝑥 minus three. And this must be equal to the original function 𝑓 sub two of 𝑥.

We’ll now subtract eight 𝑥 squared from both sides and add three, leaving us with the expression negative 𝑎𝑥 equals 𝑎𝑥. Now, in fact, we’re also told that 𝑥 is not equal to zero, so we can divide through by 𝑥 itself. And that leaves us with the equation negative 𝑎 equals 𝑎. So what values of 𝑎 make this equation true? The only way for negative 𝑎 to be equal to 𝑎 is if 𝑎 itself is equal to zero.

So, the value of 𝑎 given that 𝑓 is an even function is zero.