Consider the function 𝑓 of 𝑥 equals 𝑥 to the 𝑛 power plus 𝑏, where 𝑛 is a positive integer and 𝑏 is a constant. What value of 𝑛 would make the function linear?
So here we have our function. We know that 𝑛 is a positive integer and 𝑏 is a constant. And we want to know what value of 𝑛 would make the function linear. So to help decide what value of 𝑛 would make the function linear, let’s first understand what a positive integer is. An integer can be denoted by ℤ. The integers are negative numbers, zero, and our positive numbers. And they exclude all of the tiny decimals in between them. But we know that 𝑛 is a positive integer. So the positive integers exclude the negative numbers and zero. So the positive integers will be one, two, three, four, five, six, seven, all the way up until infinity.
So now we know what values of 𝑛 it could be. So we’re trying to decide the value of 𝑛 that would make the function linear. So a linear equation is in the form 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope. And it’s a rational number. So it could be a number that’s written as a fraction. And 𝑏 is the 𝑦-intercept. And it’s a constant. So we already have matching constants. So we need to decide what value of 𝑛 would make the function linear. So 𝑛 is the power of 𝑥.
Well, what power is 𝑥 raised to in the linear equation? Well, if nothing is there, it must be a one because 𝑥 to the first power is equal to 𝑥. And even though zero isn’t in the positive integers, let’s just look at it. 𝑥 to the zero power would actually wipe out 𝑥 because anything to the zero power is one. 𝑥 to the second power would give us 𝑥 squared, creating not a linear function but a quadratic function. And 𝑥 to the third power would create 𝑥 cubed, a cubic function. So 𝑥 to the first power is what we need.
So the value of 𝑛 that would make this function linear would be 𝑛 equals one.