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Question Video: Finding an Unknown in a Rational Function given Its Parity Mathematics • 12th Grade

Find the value of π‘Ž given 𝑓 is an even function, where 𝑓(π‘₯) = 6/(8π‘₯Β² + π‘Žπ‘₯ βˆ’ 3) and π‘₯ β‰  0.

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Video Transcript

Find the value of π‘Ž given 𝑓 is an even function, where 𝑓 of π‘₯ equals six over eight π‘₯ squared plus π‘Žπ‘₯ minus three and π‘₯ 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 π‘₯ equals 𝑓 of π‘₯ for all values of π‘₯ 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 π‘₯; it’s entirely independent of π‘₯. And so we can say that 𝑓 sub one is even.

So, for our function 𝑓 of π‘₯ 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 π‘₯ squared plus π‘Žπ‘₯ minus three. And we know that if this is even, 𝑓 sub two of negative π‘₯ must be equal to 𝑓 sub two of π‘₯. Well, 𝑓 sub two of negative π‘₯ is the expression eight times negative π‘₯ squared plus π‘Ž times negative π‘₯ minus three. That simplifies to eight π‘₯ squared minus π‘Žπ‘₯ minus three. And this must be equal to the original function 𝑓 sub two of π‘₯.

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

So, the value of π‘Ž given that 𝑓 is an even function is zero.

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