Question

question 41
which of the following equations is not a function of x?
select one
x = y^2 + 3
y = x^2 + 3
y = |x + 2| - 7
y = x^3 - 2x + 4

Explanation:

Step1: Recall the definition of a function

A relation is a function of \( x \) if for every input \( x \), there is exactly one output \( y \). We can use the vertical line test: if any vertical line intersects the graph of the relation more than once, it is not a function.

Step2: Analyze \( x = y^{2}+3 \)

Rewrite it as \( y^{2}=x - 3 \), so \( y=\pm\sqrt{x - 3} \). For a given \( x>3 \), there are two values of \( y \) (positive and negative square root). So this fails the vertical line test.

Step3: Analyze \( y = x^{2}+3 \)

For any real \( x \), plugging in \( x \) gives exactly one \( y \) (since squaring a number and adding 3 gives a unique result). Passes vertical line test.

Step4: Analyze \( y=\vert x + 2\vert-7 \)

The absolute value function gives a unique \( y \) for each \( x \) (the absolute value of a number is unique, then subtract 7). Passes vertical line test.

Step5: Analyze \( y=x^{3}-2x + 4 \)

A cubic function is a polynomial, and for any real \( x \), there is exactly one \( y \) (polynomial functions are functions). Passes vertical line test.

Answer:

\( \boldsymbol{x = y^{2}+3} \) (the first option)