Determine whether the function is a one-to-one function.
-$$y = x ^ { 2 } + 10 x + 29 , x \geq - 5$$
A)Yes
B)No

Question

Quizplus · Accepted Answer

A function is one-to-one if it passes both the vertical line test (which all functions do) and the horizontal line test, meaning no two different inputs (x-values) produce the same output (y-value). The given function $y = x^2 + 10x + 29$ is a quadratic function, which generally is not one-to-one over its entire domain because it is a parabola. However, the domain of this function is restricted to $x \geq -5$. The vertex of this parabola, found by completing the square or using the vertex formula $x = -\frac{b}{2a}$, is at $x = -5$. Since the parabola opens upwards and the domain starts at the vertex and extends to the right, the function is increasing over its entire domain. Therefore, it will pass the horizontal line test in this restricted domain, making it a one-to-one function.

Determine Whether the Function Is a One-To-One Function y=x2+10x+29,x≥−5y = x ^ { 2 } + 10 x + 29 , x \geq - 5y=x2+10x+29,x≥−5

Determine Whether the Function Is a One-To-One Function $y = x ^ { 2 } + 10 x + 29 , x \geq - 5$