To determine the type of the function [tex]\( f(x) = 2x^3 - 4x^2 + 5 \)[/tex], we will analyze the form and components of the function.
1. Polynomial Function: A polynomial function is an expression that includes variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. It generally takes the form:
[tex]\[
f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0,
\][/tex]
where [tex]\( a_n, a_{n-1}, \dots, a_0 \)[/tex] are constants, and [tex]\( n \)[/tex] is a non-negative integer.
2. Exponential Function: An exponential function has the variable in the exponent and usually takes the form:
[tex]\[
f(x) = a \cdot b^x,
\][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants and [tex]\( b \)[/tex] is a positive real number not equal to 1.
3. Logarithmic Function: A logarithmic function is the inverse of an exponential function and takes the form:
[tex]\[
f(x) = a \cdot \log_b(x),
\][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants, and [tex]\( b \)[/tex] is the base of the logarithm.
4. Radical Function: A radical function includes a root, and it typically appears in the form:
[tex]\[
f(x) = a \cdot \sqrt[n]{x},
\][/tex]
where [tex]\( a \)[/tex] is a constant and [tex]\( n \)[/tex] is an integer greater than or equal to 2.
Now, let's identify the type of the given function [tex]\( f(x) = 2x^3 - 4x^2 + 5 \)[/tex].
- The function has terms involving [tex]\( x \)[/tex] raised to the power of 3 and 2, both of which are non-negative integers.
- There are no exponents where [tex]\( x \)[/tex] is in the exponent position.
- There are no logarithmic terms.
- There are no roots involved.
Since the function [tex]\( f(x) = 2x^3 - 4x^2 + 5 \)[/tex] is composed of terms where [tex]\( x \)[/tex] is raised to non-negative integer powers, it fits the form of a polynomial function.
Therefore, the type of function is Polynomial.
The answer is Polynomial.
