Question

Find the following for each function: (a) \(f(0)\) (b) \(f(1)\) (c) \(f(-1)\) (d) \(f(-x)\) (e) \(-f(x)\) (f) \(f(x+1)\) (g) \(f(2 x)\) (h) \(f(x+h)\) \(f(x)=1-\frac{1}{(x+2)^{2}}\)

Short answer

(a) \(\frac{3}{4}\), (b) \(\frac{8}{9}\), (c) 0, (d) \(1 - \frac{1}{(-x+2)^2}\), (e) \(-1 + \frac{1}{(x+2)^2}\), (f) \(1 - \frac{1}{(x+3)^2}\), (g) \(1 - \frac{1}{4(x+1)^2}\), (h) \(1 - \frac{1}{(x+h+2)^2}\)

Step-by-step solution

Evaluate f(0)

To find the value of the function at 0, substitute 0 for x in the function: \[ f(0) = 1 - \frac{1}{(0+2)^2} = 1 - \frac{1}{4} = \frac{3}{4} \]

Evaluate f(1)

To find the value of the function at 1, substitute 1 for x in the function: \[ f(1) = 1 - \frac{1}{(1+2)^2} = 1 - \frac{1}{9} = \frac{8}{9} \]

Evaluate f(-1)

To find the value of the function at -1, substitute -1 for x in the function: \[ f(-1) = 1 - \frac{1}{(-1+2)^2} = 1 - \frac{1}{1} = 0 \]

Evaluate f(-x)

To find the expression for f(-x), substitute -x for x in the function: \[ f(-x) = 1 - \frac{1}{(-x+2)^2} \]

Evaluate -f(x)

To find the expression for -f(x), multiply the function by -1: \[ -f(x) = - \left(1 - \frac{1}{(x+2)^2} \right) = -1 + \frac{1}{(x+2)^2} \]

Evaluate f(x+1)

To find the expression for f(x+1), substitute x+1 for x in the function: \[ f(x+1) = 1 - \frac{1}{(x+1+2)^2} = 1 - \frac{1}{(x+3)^2} \]

Evaluate f(2x)

To find the expression for f(2x), substitute 2x for x in the function: \[ f(2x) = 1 - \frac{1}{(2x+2)^2} = 1 - \frac{1}{4(x+1)^2} \]

Evaluate f(x+h)

To find the expression for f(x+h), substitute x+h for x in the function: \[ f(x+h) = 1 - \frac{1}{(x+h+2)^2} \]

Key concepts

Function Evaluation

First of all, let's understand what function evaluation is. Function evaluation means finding the value of a function when an input value is given for the variable(s). In simple terms, you plug in a value for the variable, typically represented by 'x', into the function and solve. For example, when we want to evaluate the function given as \(f(x)=1-\frac{1}{(x+2)^{2}}\), we substitute different values for x like 0, 1, or any other number given in an exercise. Evaluating functions helps in understanding how the function behaves with various inputs.

Substitution in Functions

Substitution in functions is when you replace the variable (often 'x') with a specific value or another expression. Consider our function again: \(f(x)=1-\frac{1}{(x+2)^{2}}\). To find \(f(0)\), we substitute 0 for every 'x' in the function: \( f(0) = 1 - \frac{1}{(0 + 2)^2} \) Similarly, for other values like 1 or -1, we do the same substitution. It is a crucial step to find the exact value of the function for any input.Here are other examples:

• For \( f(-x) \), replace every 'x' with '-x'.
• For \( f(2x) \), replace 'x' with '2x'.
• For \( f(x+1) \), replace 'x' with 'x+1'.

Substitution is very straightforward, but it's essential to do it carefully to avoid mistakes.

Algebraic Manipulation

Algebraic manipulation involves simplifying expressions, combining like terms, and other algebra operations. Let's take a look at an example from our function: When we substituted 2x into our function: \( f(2x) = 1 - \frac{1}{(2x + 2)^2} \) We can further simplify this to make it more concise: \( f(2x) = 1 - \frac{1}{4(x + 1)^2} \) In algebra, it's about making things as simple as can be. These manipulations help us understand the function's behavior without extra complexities. Remember to always follow the order of operations (PEMDAS/BODMAS) to simplify terms correctly.

Composite Functions

Composite functions are functions nested inside each other. This means one function's output becomes another function's input. For example, if we have two functions, \( f \) and \( g \), the composite function \( f(g(x)) \) means you first apply \( g \) and then \( f \). With our example function, if you needed to find \( f(f(x+1)) \), you would first solve for \( f(x+1) \) as: \( f(x+1) = 1 - \frac{1}{(x+3)^2} \) Then you use the result in the function again. Composite functions help compound effects, and they require careful following through each function independently. When evaluating composite functions, ensure you correctly apply each function step by step to avoid missed calculations and errors.