khan academy absolute value inequalities

This makes sense because x 4 can't be negative, and because (1 – x) 2 can't be negative. Thus, | f( x)| is not always the same as f( x).īy process of elimination, the answer is f( x) = x 4 + (1 – x) 2. For example, let's find f( x) when x = –1/2. If we let x be a small negative fraction, then x 4 would be close to zero, and we would be left with x, which is negative. Therefore, x 4 cannot be negative, because if we multiplied a negative number by itself four times, the result would be positive. In general, any number taken to an even-numbered power must always be non-negative. Thus, there are some values on f( x) that are negative, so we can eliminate this function. Therefore, let's evaluate f( x) when x is a fractional value such as 1/2. However, if x 2 is really small, then adding –2 x could make it negative.

We know that x 2 by itself can never be negative. Thus | f( x)| will not be the same as f( x), and we can eliminate this choice as well. If we let x = 1, then f(1) = 1 – 9 = –8, which is negative.

Therefore, f( x) = 2 x + 3 isn't the correct answer. Thus, f( x) has negative values, and if we were to graph | f( x)|, the result would be different from f( x). Obviously, this equation of a line will have negative values. In other words, our answer will consist of the function that never has negative values. If we can show that f( x) has negative values, then | f( x)| will be different from f( x) at some points, because its negative values will be changed to positive values.

However, any values of f( x) that are positive or equal to zero will not be changed, because the absolute value of a positive number (or zero) is still the same number. Essentially, | f( x)| will take all of the negative values of f( x) and reflect them across the x-axis. When we take the absolute value of a function, any negative values get changed into positive values.

Khan academy absolute value inequalities