Problem 11 - Algebra A Combined Function

Question

If \(f(x)=x^{2}-6 x+2, g(x)=-2 x\), and \(h(x)=\sqrt{x}\), find each composition. $$ (g \circ f)(-1) $$

Step-by-Step Solution

Answer

The composition \((g \circ f)(-1)\) is \(-18\).

Step 1: Understand the problem

We need to find the composition of two functions, \(g\) and \(f\), specifically \((g \circ f)(-1)\). This means we will first find \(f(-1)\) and then use this result as the input for \(g\).

Step 2: Evaluate \(f(-1)\)

Substitute \(-1\) into the function \(f(x) = x^2 - 6x + 2\):\[ f(-1) = (-1)^2 - 6(-1) + 2 \]Simplify:\[ f(-1) = 1 + 6 + 2 = 9 \].

Step 3: Evaluate \(g(f(-1))\)

Now substitute \(f(-1) = 9\) into the function \(g(x) = -2x\):\[ g(9) = -2(9) \]Simplify:\[ g(9) = -18 \].

Step 4: Conclusion

The composition \((g \circ f)(-1)\) evaluates to \(-18\).

Key Concepts

Evaluating Functions

Evaluating functions is an essential skill in mathematics that involves substituting values into a function to find the output or result. This process helps us make predictions and understand relationships between variables. For example, with the function \( f(x) = x^2 - 6x + 2 \), you can evaluate the function at any value of \( x \) by replacing \( x \) with that value.

• Start by identifying which function you want to evaluate. In our exercise, we first need \( f(-1) \).
• Substitute the value into the function. For \( f(x) = x^2 - 6x + 2 \) and \( x = -1 \), this becomes \( (-1)^2 - 6(-1) + 2 \).
• Simplify the expression step by step: \( 1 + 6 + 2 = 9 \), so \( f(-1) = 9 \).

By evaluating \( f(-1) \), we use this result to continue solving problems, like in function composition.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically written in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \( a \).Here are key features of quadratic functions:

• The vertex represents the highest or lowest point. It can be found with the formula \( x = -\frac{b}{2a} \).
• The axis of symmetry is a vertical line through the vertex, given by \( x = -\frac{b}{2a} \).
• Roots or zeros of a quadratic function occur where the graph intersects the x-axis. They can be found using factoring, completing the square, or the quadratic formula.

Understanding quadratic functions involves recognizing their standard form and interpreting their graphs. They are widely used in various applications, from physics to finance, due to their predictable patterns and behavior.

Function Notation

Function notation is a convenient way to express relationships where each input has a single output. It's common in mathematics to use letters like \( f \), \( g \), or \( h \) to denote functions. When using function notation:

• \( f(x) \) means "the function \( f \) evaluated at \( x \)." Here, \( x \) is the input variable.
• Functions can have different notations, like \( f(x) = x^2 - 6x + 2 \) as given in our task.
• We can construct new functions by composing two or more functions. This involves inserting one function's result into another, noted as \((g \circ f)(x)\). For example, if \( g(x) = -2x \), we would perform \((g \circ f)(-1)\) by finding \( f(-1) \) first, then using that result in \( g \).

Using function notation helps streamline complex mathematical expressions and allows us to communicate ideas more clearly and efficiently.