## 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:

$\fn_jvn {\color{DarkBlue} y=-3x^4+5}$

## 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, $\fn_jvn -3x^4$. 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 $\inline \fn_jvn 3x^4$ is zero — that happens where $\inline \fn_jvn x=0$ — and there is no limit to how high $\inline \fn_jvn 3x^4$ can go, so $\inline \fn_jvn 3x^4$ ranges from 0 to infinity. Then its negative, $\fn_jvn -3x^4$ ranges from negative infinity to 0.

 $\fn_jvn y=x^4$$\fn_jvn y=x^4$ $\fn_jvn y=-x^4$$\fn_jvn y=-x^4$ $\fn_jvn y=-x^4+5$$\fn_jvn y=-x^4+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, $\fn_jvn 0+5=5$, so the top of the range is 5.

So the range of $\inline \fn_jvn y=-3x^4+5$ is from negative infinity to 5. In other words, $\inline \fn_jvn y\leq 5$. In interval notation, that’s $\inline \fn_jvn \left ( -\infty , 5 \right )$.

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