To determine which function has the domain \({x : x \in \mathbb{R}, x \neq \frac{2}{7}}\), we should examine each option for any restrictions on the input \(x\).
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A. \(V(x) = \frac{1}{2x - 7}\)
The function is undefined when the denominator is zero: \[ 2x - 7 = 0 \implies 2x = 7 \implies x = \frac{7}{2} \] Hence, the domain is all real numbers except \(x = \frac{7}{2}\). -
B. \(U(x) = \frac{1}{7x + 2}\)
This function is undefined when the denominator is zero: \[ 7x + 2 = 0 \implies 7x = -2 \implies x = -\frac{2}{7} \] Thus, the domain is all real numbers except \(x = -\frac{2}{7}\). -
C. \(T(x) = \frac{1}{7x - 2}\)
This function is undefined when the denominator is zero: \[ 7x - 2 = 0 \implies 7x = 2 \implies x = \frac{2}{7} \] Therefore, the domain is all real numbers except \(x = \frac{2}{7}\). -
D. \(Q(x) = \frac{1}{2x - 7}\)
Similar to option A, this function is undefined when the denominator is zero: \[ 2x - 7 = 0 \implies 2x = 7 \implies x = \frac{7}{2} \] Thus, the domain is all real numbers except \(x = \frac{7}{2}\).
The only function from the options provided that has the specified domain \({x : x \in \mathbb{R}, x \neq \frac{2}{7}}\) is:
C. \(T(x) = \frac{1}{7x - 2}\).