Questions: Which of the following functions is not a one-to-one function? f(x)=x g(x)=2x+1 h(x)=-2x^2+2 j(x)=5x^3+4 k(x)=3x^5-4

Solution Steps

To determine which function is not a one-to-one function, we need to check if each function passes the horizontal line test. A function is one-to-one if and only if every horizontal line intersects the graph of the function at most once.

We are given the following functions: \[ \begin{align_} f(x) &= x \\ g(x) &= 2x + 1 \\ h(x) &= 2 - 2x^2 \\ j(x) &= 5x^3 + 4 \\ k(x) &= 3x^5 - 4 \end{align_} \]

A function is one-to-one if it passes the horizontal line test, which means that every horizontal line intersects the graph of the function at most once. This can be checked by examining the derivative of each function. If the derivative is always positive or always negative, the function is one-to-one.

\[ f'(x) = 1 \] The derivative is always positive, so \( f(x) \) is one-to-one.

\[ g'(x) = 2 \] The derivative is always positive, so \( g(x) \) is one-to-one.

\[ h'(x) = -4x \] The derivative is zero at \( x = 0 \) and changes sign around this point, so \( h(x) \) is not one-to-one.

\[ j'(x) = 15x^2 \] The derivative is zero at \( x = 0 \) and positive elsewhere, so \( j(x) \) is not one-to-one.

\[ k'(x) = 15x^4 \] The derivative is zero at \( x = 0 \) and positive elsewhere, so \( k(x) \) is not one-to-one.

Final Answer