91Ó°ÊÓ

Function evaluation is the process of determining the output of a function for a particular input. To evaluate a function, you replace its variable with a given number or expression. In the context of our exercise, the function given was
\( f(x) = \sqrt{x+3} - x + 1 \).
To find
\( f(1) \),
we substituted
\( x \)
with
\( 1 \). This substitution resulted in
\( f(1) = \sqrt{1+3} - 1 + 1 \). After simplification, the square root of 4, which is 2, combined with the remaining terms led to the final answer of 2. It's crucial for students to understand that function evaluation often requires simplifying complex expressions and diligently following the order of operations to obtain the correct result.

Radical expressions include numbers or expressions under a root symbol, such as square roots or cube roots. In our function
\( f(x) = \sqrt{x+3} - x + 1 \),
the expression
\( \sqrt{x+3} \)
is a radical expression because it contains a square root. Handling radical expressions can involve operations such as simplification, addition, subtraction, multiplication, or division. When we evaluated \( f(1) \), we simplified the radical expression
\( \sqrt{1+3} = \sqrt{4} \)
to 2. This simplification of radicals requires students to identify perfect squares, cubes, etc., or to break down the expression into smaller radical components if it's not a perfect power.

Polynomial functions involve terms with variables raised to whole number exponents and their coefficients. In our example, the functions
\( g(t) = t^2 - 1 \)
and
\( h(x) = x^2 + \frac{1}{x} + 2 \)
are both polynomial functions. The first function is a quadratic polynomial, expressed by
\( t^2 - 1 \)
, which has a degree of two. The second function is more complex with a leading term of
\( x^2 \)
, which determines it as a quadratic function, but it also contains
\( \frac{1}{x} \), a rational term. While it is not a polynomial term, in this context, it doesn't change the quadratic nature of the function. In evaluating polynomial functions, one must apply exponent rules, perform polynomial addition and subtraction, and simplify where possible. For instance, if we were to evaluate \( g(t) \) at \( t = 1 \), we would calculate \( 1^2 - 1 = 0 \).