Evaluate each function at the given values of the...

Evaluate each function at the given values of the independent variable and simplify.
$g(x)=x^{2}-10 x-3$
$\begin{array}{lll}\text { a. } g(-1) & \text { b. } g(x+2) & \text { c. } g(-x)\end{array}$

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Key Concepts

Function Evaluation

Function evaluation is the process of determining the output of a function by substituting a specified value or expression in place of the variable. This concept is fundamental in analyzing how functions behave under different inputs.

Substitution

Substitution involves replacing the variable in a function with another value or expression. This technique is crucial when calculating the function's output at transformed inputs or when performing composite function evaluations.

Polynomial Simplification

Polynomial simplification consists of expanding, factoring, and combining like terms to rewrite a polynomial in its simplest form. Mastery of this process is essential when evaluating functions, as it improves clarity and accuracy of the computed result.

Transcript

00:01 For this problem, we're asked to evaluate this function, g of x is equal to x squared minus 10x minus 3 at each given value of the independent variable and then to simplify.

00:13 So there's three different parts of this problem, so we'll begin with part a, which asks us to find g of negative 1.

00:20 So essentially all we're doing here is plugging in negative 1 for every x in this function.

00:27 So we have x squared to begin, so we'll have negative 1 squared, minus 10 times x so 10 times negative 1 and then minus 3.

00:39 Now if we simplify this, negative 1 squared is positive 1.

00:44 Negative 10 times negative 10 is positive 10.

00:48 And then we have this negative 3.

00:51 Now 1 plus 10 is 11 and then 11 minus 3 is equal to 8.

01:00 So our answer for part a is 8.

01:05 Now for part b, we asked to find g of x plus 2.

01:10 So now let's plug x plus 2 in for every x value.

01:14 So we'll have x plus 2 squared minus 10 times x plus 2 minus 3.

01:24 So since we have this binomial x plus 2 to the second power, we can write that out as x plus 2 times x plus 2 and then we can use the distributive property to multiply this out.

01:37 So x times x is x squared, 2 times x is 2x, x times 2x, and 2 times 2x, and 2 times 2 is 4.

01:51 Now what we have up here next is negative 10 times x plus 2, so let's distribute that 10 in...

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