What is the range of f(x) = 2x^2 + 7?
Understand the Problem
The question asks to determine the range of the function f(x) = 2x^2 + 7. Since x^2 is always non-negative, 2x^2 will also be non-negative. Therefore, the minimum value of f(x) occurs when x = 0. We need to find this minimum value and express the range accordingly.
Answer
$f(x) \ge 7$
Answer for screen readers
$f(x) \ge 7$
Steps to Solve
- Find the minimum value of $2x^2$
Since $x^2$ is always greater than or equal to 0 for any real number $x$, the smallest value $x^2$ can take is 0. Therefore, the smallest value of $2x^2$ is $2 * 0 = 0$.
- Find the minimum value of $f(x)$
The minimum value of $f(x) = 2x^2 + 7$ occurs when $2x^2$ is at its minimum, which is 0. Thus, the minimum value of $f(x)$ is $0 + 7 = 7$.
- Determine the range
Since $2x^2$ is always non-negative, $f(x) = 2x^2 + 7$ will always be greater than or equal to 7. Therefore, the range of $f(x)$ is all values greater than or equal to 7.
$f(x) \ge 7$
More Information
The range represents all possible output values of the function. In this case, since the parabola opens upwards and its vertex is at (0, 7), the function will always output values greater than or equal to 7.
Tips
A common mistake is to forget that $x^2$ is always non-negative. Another mistake is to state the range as all real numbers, not recognizing the lower bound imposed by the $+7$ and the $2x^2$ term.
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