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Finding a Function’s Range

This post about finding a function’s range from its equation is part of a series of posts to help you prepare for the Advanced Algebra and Functions part of the Accuplacer test.

Question

Find the range of this function:


Solution

A function’s range is the set of values y can take in that function.

You are asked to find a function’s range. What are the values y can take in this function?

Consider the first term, . Any real number (unless otherwise stated, it’s safe to assume x is real) to an even power, like 4, has a positive value or zero. It just can’t be negative. The lowest value of is zero — that happens where  — and there is no limit to how high can go, so  ranges from 0 to infinity. Then its negative,  ranges from negative infinity to 0.

Graph of y = x to the fourth power. The graph extends from 0 to infinity. graph of y=negative x to the fourth power. The graph extends from negative infinity to 0. Graph of y = negative 3 x to the fourth power plus 5. Graph of y = x^4. The graph extends from negative infinity to 5.

The second term, 5. increases each y-value by 5. At the low end of the range, negative infinity plus 5 is still negative infinity, so the bottom of the range is still negative infinity. At the top end, , so the top of the range is 5.

So the range of is from negative infinity to 5. In other words, . In interval notation, that’s .

This question is similar to question number 11 in the sample questions for the Accuplacer Advanced Algebra and Functions test.