Example Question - determining domain
Here are examples of questions we've helped users solve.
Determining the Domain of a Function
The function provided in the image is: g(x) = 5 * 3^x To find the domain of the function, we need to determine the set of all possible input values (x) that the function can accept without causing any mathematical issues, such as division by zero or taking the square root of a negative number. In this case, the function g(x) consists of a constant 5 multiplied by 3 raised to the power of x. For any real number x, 3^x will yield a real number, and multiplying by 5 will also result in a real number. There are no restrictions due to division by zero or square roots (or any other operation that could restrict the domain). Therefore, the domain of the function g(x) is all real numbers, which is often denoted as (-∞, ∞) or simply ℝ.
Determining the Domain of a Function
The function provided is: \[ f(x) = \frac{1}{6x + 7} \] To find the domain of this function, we must determine the values of x for which the function is defined. Since we have a fraction with a denominator of \(6x + 7\), we must ensure that the denominator is never 0, because division by zero is undefined. So we set up the inequality: \[ 6x + 7 \neq 0 \] Solving for x, we subtract 7 from both sides: \[ 6x \neq -7 \] And then divide by 6: \[ x \neq \frac{-7}{6} \] Thus, the only value that x cannot take (the only value not in the domain) is \(-\frac{7}{6}\).
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