Which equation is not a linear function?
1 answer:
<h3>Answer:</h3>
a, e
<h3>Explanation:</h3>If any of the terms has a variable with an exponent other than 1 or 0, or has a sum of variable exponents other than 1, the term is non-linear and the relation is not a linear relation.
a: the term xy has a sum of exponents of 1+1=2, so is not a linear term.
e: the term x² has an exponent other than 0 or 1, so is not a linear term.
In the attached graph, the non-linear relations are shown graphed in black. The remaining relations are all linear.
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<em>Comment on linear function</em>
By one definition, a linear function is one that is of the form
... f(x) = ax + b
Here, an equation such as <em>y = x + 2y</em> can be put in that form, but it is <em>not</em> in that form as presented. Yes, the graph is of a straight line, but you would have a hard time identifying independent and dependent variables from the equation as given.
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(tanθ + cotθ)² = sec²θ + csc²θ
<u>Expand left side</u>: tan²θ + 2tanθcotθ + cot²θ
<u>Evaluate middle term</u>: 2tanθcotθ = = 2
⇒ tan²θ + 2+ cot²θ
= tan²θ + 1 + 1 + cot²θ
<u>Apply trig identity:</u> tan²θ + 1 = sec²θ
⇒ sec²θ + 1 + cot²θ
<u>Apply trig identity:</u> 1 + cot²θ = csc²θ
⇒ sec²θ + csc²θ
Left side equals Right side so equation is verified