Question

Assume \(f\) is an even function and \(g\) is an odd function. Use the table to evaluate the given compositions. $$\begin{array}{lrrrr} \hline x & 1 & 2 & 3 & 4 \\ f(x) & 2 & -1 & 3 & -4 \\ g(x) & -3 & -1 & -4 & -2 \\ \hline \end{array}$$ a. \(f(g(-1))\) b. \(g(f(-4))\) c. \(f(g(-3))\) d. \(f(g(-2))\) e. \(g(g(-1))\) f. \(f(g(0)-1)\) g. \(f(g(g(-2)))\) h. \(g(f(f(-4)))\) i. \(\quad g(g(g(-1)))\)

Short answer

Question: Evaluate the following compositions using the given table and properties of odd and even functions: \(f(g(-1))\), \(g(f(-4))\), \(f(g(-3))\), \(f(g(-2))\), \(g(g(-1))\), \(f(g(0)-1)\), \(f(g(g(-2)))\), \(g(f(f(-4)))\), and \(g(g(g(-1)))\). Answer: 1. \(f(g(-1)) = 3\) 2. \(g(f(-4)) = -2\) 3. \(f(g(-3)) = -4\) 4. \(f(g(-2)) = 2\) 5. \(g(g(-1)) = -4\) 6. \(f(g(0)-1) = 2\) 7. \(f(g(g(-2))) = 3\) 8. \(g(f(f(-4))) = -2\) 9. \(g(g(g(-1))) = -2\)

Step-by-step solution

Recall Properties of Even and Odd Functions

An even function is a function that satisfies \(f(-x) = f(x)\) for all \(x\). An odd function is a function that satisfies \(g(-x) = -g(x)\) for all \(x\). These properties will be helpful for evaluating the compositions.

Evaluate \(f(g(-1))\)

Apply the property of the odd function to \(g(-1)\): $$g(-(-1)) = g(1) = -3$$Now substitute this value into the even function: $$f(g(-1)) = f(-3) = 3$$ The requested value is 3.

Evaluate \(g(f(-4))\)

Apply the property of the even function to \(f(-4)\): $$f(-(-4)) = f(4) = -4$$Now substitute this value into the odd function: $$g(f(-4)) = g(-4) = -2$$ The requested value is -2.

Evaluate \(f(g(-3))\)

Apply the property of the odd function to \(g(-3)\): $$g(-(-3)) = g(3) = -4$$Now substitute this value into the even function: $$f(g(-3)) = f(-4) = -4$$ The requested value is -4.

Evaluate \(f(g(-2))\)

Apply the property of the odd function to \(g(-2)\): $$g(-(-2)) = g(2) = -1$$Now substitute this value into the even function: $$f(g(-2)) = f(-1) = 2$$ The requested value is 2.

Evaluate \(g(g(-1))\)

Apply the property of the odd function to \(g(-1)\): $$g(-(-1)) = g(1) = -3$$Then evaluate the composition using the table: $$g(g(-1)) = g(-3) = -4$$ The requested value is -4.

Evaluate \(f(g(0)-1)\)

Evaluate \(g(0)\) using the property of odd functions: $$g(-0) = -g(0) = 0$$As a result, we get: $$f(g(0)-1) = f(0-1) = f(-1) = 2$$ The requested value is 2.

Evaluate \(f(g(g(-2)))\)

Apply the property of the odd function to find \(g(-2)\): $$g(-(-2)) = g(2)=-1$$ Now evaluate the composition using the table: $$g(g(-2)) = g(-1) = -3$$Finally, substitute this value into the even function: $$f(g(g(-2))) = f(-3) = 3$$ The requested value is 3.

Evaluate \(g(f(f(-4)))\)

Apply the property of the even function to find \(f(-4)\): $$f(4) = -4$$ Now evaluate the composition using the table: $$f(f(-4)) = f(-4) = -4$$Finally, substitute this value into the odd function: $$g(f(f(-4))) = g(-4) = -2$$ The requested value is -2.

Evaluate \(g(g(g(-1)))\)

Apply the property of the odd function to find the first composition of \(g(-1)\): $$g(-(-1)) = g(1) = -3$$ Now, apply the property again to find the second composition: $$g(g(-1)) = g(-3) = -4$$Finally, evaluate the last composition using the table: $$g(g(g(-1))) = g(-4) = -2$$ The requested value is -2.

Key concepts

Function Composition

Function composition is like putting one function inside another. You take the output of one function and make it the input of another. Imagine playing with colored blocks as a child. If you stack a blue block on top of a red block, you're creating a new, multi-colored tower. This is similar to function composition. You take function \(f(x)\) and function \(g(x)\), and create a new function \((f \circ g)(x)\), defined as \(f(g(x))\).
The order of composition matters a lot. \(f(g(x))\) is usually not the same as \(g(f(x))\), much like putting jam on bread is not the same as putting bread on jam! When solving exercises involving function composition, always track which function is applied first. Identify the innermost function (like \(g(x)\)) and solve it before applying the outer function (\(f(x)\)). This approach ensures that each step builds upon the previous one, much like adding more blocks to your tower.

Properties of Functions

Understanding properties of functions, such as even and odd properties, adds an exciting layer to solving problems. For even functions \(f\), you have the useful rule: \(f(-x) = f(x)\). This means that the function's value remains the same even if the input's sign changes. Imagine a perfectly symmetrical pattern that looks identical when flipped.
On the other hand, odd functions \(g\) have a different rule: \(g(-x) = -g(x)\). This property is like a seesaw perfectly balancing opposites; flipping the input flips the output sign too. In problem-solving, these properties make it easier to compute values without recalibrating for each change of sign. Always applying these rules allows for quick evaluations of expressions without directly substituting every possible value. This approach shows how powerful properties can streamline complex mathematical tasks.

Function Evaluation

Function evaluation involves plugging specific numbers into an equation and finding the result. It's like cooking following a recipe where you measure out exact ingredients. In the given exercise with functions \(f\) and \(g\), evaluation uses a table to find outputs for certain inputs.
Evaluating \(f(x)\) means taking a number, finding where it fits in \(f\)'s row, and reading off the result. The same process applies to \(g(x)\). This method shows how functions work like machines—input a number (ingredient), and get a result (dish).

• If it’s an even function like \(f\), knowing that \(f(-x) = f(x)\) simplifies evaluations.
• For odd functions like \(g\), \(g(-x) = -g(x)\) helps quickly determine outputs even with negative inputs.
• Function composition compounds this by running one function’s output through another, simplifying through known properties.

Function evaluation is essential for mathematical investigation and provides vital insights when partnered with properties and composition.