Which of the following statements is true about the function y = 5⋅x4 ?
It is an even function
It is an odd function
It is an exponential function
It is an inverse function
To determine which of the following statements is true about the function y = 5⋅x^4, we first need to understand the definitions of the given terms:
1. Even function: An even function is symmetric about the y-axis, which means that if you mirror the graph across the y-axis, the graph remains the same. In an even function, if you replace x with -x, the function will still have the same output value.
2. Odd function: An odd function is symmetric about the origin (0,0), which means that if you rotate the graph 180 degrees around the origin, the graph remains the same. In an odd function, if you replace x with -x, the function will have the same output value but with the opposite sign.
3. Exponential function: An exponential function is a function in which the variable appears in an exponent.
4. Inverse function: An inverse function is a function that "undoes" the original function. If f(x) is the original function, then the inverse function is denoted as f^(-1)(x), and it has the property that f(f^(-1)(x)) = x for all x in the domain of f(x).
Now, let's analyze the given function y = 5⋅x^4 to determine which statement is true.
1. Even function: To determine if a function is even, we need to check if f(-x) = f(x) for all x in the domain. Let's substitute -x into the function:
f(-x) = 5⋅(-x)^4 = 5⋅x^4
Since f(-x) = f(x), the function is even. So the statement "It is an even function" is true.
2. Odd function: To determine if a function is odd, we need to check if f(-x) = -f(x) for all x in the domain. Let's substitute -x into the function:
f(-x) = 5⋅(-x)^4 = 5⋅x^4
Since f(-x) ≠ -f(x), the function is not odd. So the statement "It is an odd function" is false.
3. Exponential function: An exponential function is a function in which the variable appears in an exponent. In the given function y = 5⋅x^4, the variable x is raised to the power of 4 (x^4). Thus, the statement "It is an exponential function" is false.
4. Inverse function: To determine if a function has an inverse, we need to check if it passes the horizontal line test, which means that every horizontal line intersects the graph at most once. Since the given function y = 5⋅x^4 is continuous and passes the horizontal line test, it has an inverse function. However, the given function itself is not an inverse function. Therefore, the statement "It is an inverse function" is false.
In summary, the correct statement is: "It is an even function."