Question

the function p (not shown) is a polynomial function of degree 3. the graphs of four functions f, g, h , and k are given. the output values of p are the same as the output values of the composition function when p is composed with one of these functinos as the input function. for which of the functions is this true?

Answer 1

To find which function results in the same output values as the polynomial function p when composed, we need to find the inverse equation of p and compare it to the given functions.

Step-by-step explanation:

The question is asking which of the given functions, f, g, h, or k, results in the same output values as the polynomial function p when composed. In other words, when we substitute the output of any of these functions as the input for p, we get the same output value. To determine this, we need to find the function that is an inverse of p.

To find the inverse of p, we need to switch the x and y variables in the equation. Then, solve the equation for the new y variable. If any of the given functions matches this inverse equation, then it is the correct answer.

Once we identify the inverse of p, we can determine which of the given functions matches that inverse equation, revealing which function results in the same output values as p when composed.

Answer 2

The function for which the output values of
`\( p \)` are the same as the output values of the composition function is
`\( f \)`.

Step-by-step explanation:

To determine which function among
`\( f, g, h, \)` and
`\( k \)` has the same output values as the composition function with
`\( p \)`, we need to analyze the graphs and the nature of polynomial functions. Since
`\( p \)` is a polynomial function of degree 3, its composition with another function results in a polynomial function of the same degree.

Looking at the options,
`\( f \)` is the only function with a cubic (degree 3) graph. When
`\( p \)` is composed with
`\( f \)`, the resulting function maintains a degree of 3. This implies that the output values of
`\( p \)` and the composition function are the same, making
`\( f \)` the correct choice.

In mathematical terms, if
`\( p(x) \)` is a cubic polynomial and
`\( f(x) \)` is also a cubic function, then
`\( (p \circ f)(x) \)` will be a cubic polynomial as well, preserving the degree of the original polynomial. Therefore, the graph of
`\( f \)` aligns with the characteristics of
`\( p \)`, making
`\( f \)` the function for which the statement is true.