Functions and Graphs Exercise

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Questions and Answers

If $f(x) = x^2 - x$, what is $f(-2)$?

If $f(x) = rac{h}{6x - 9}$, simplify $f(a + h) - f(a)$.

$\frac{h}{6(a+h)-9} - \frac{h}{6a-9}$

Express the perimeter ($P$) of a square as a function of its area ($A$).

$P(A) = 4\sqrt{A}$

What is the domain of $g(x) = \sqrt{x^2 - 4}$?

$(-∞, -2] ∪ [2, ∞)$ Signup and view all the answers

Simplify $f(x) = x^2 + 2x^2 - 1$.

$3x^2 - 1$ Signup and view all the answers

Given the functions $f(x) = 2x + 1$ and $g(x) = x^3 + 1$, find $(f ext{ o } g)(x)$.

$f(g(x)) = 2(x^3 + 1) + 1 = 2x^3 + 3$ Signup and view all the answers

Given the functions $f(x) = sqrt{x} + 1$ and $g(x) = \frac{1}{x}$ for $x \ne 0$, find $(g ext{ o } f)(x)$.

$g(f(x)) = \frac{1}{\sqrt{x} + 1}$ Signup and view all the answers

For $f(x) = \frac{1}{\sqrt{x} - 1}$ where $x \ge 1$, find $(f ext{ o } f)(x)$.

$f(f(x)) = \frac{1}{\sqrt{\frac{1}{\sqrt{x} - 1}} - 1}$ Signup and view all the answers

Given the function $f(x) = -2x + 8$, find $f^{-1}(x)$.

$f^{-1}(x) = \frac{8 - x}{2}$ Signup and view all the answers

For the function $f(x) = 3x^3 + 7$, verify $f(f^{-1}(x)) = f^{-1}(f(x)) = x$.

First, find $f^{-1}(x) = \sqrt[3]{\frac{x - 7}{3}}, then: \ f(f^{-1}(x)) = 3(\sqrt[3]{\frac{x - 7}{3}})^3 + 7 = x \ and \ f^{-1}(f(x)) = \sqrt[3]{\frac{3x^3 + 7 - 7}{3}} = x$ Signup and view all the answers

What is the value of $g(x)$ when $x = -3$ for the function defined as $g(x) = \begin{cases} 6x+7, & x \leq -2 \ 4-3, & -2 < x \leq 1 \ -x+1, & x > 1 \end{cases}$?

g(x) = -11 Signup and view all the answers

For $g(x) = \begin{cases} 6x + 7, & x \leq -2 \ 4-3, & -2 < x \leq 1 \ -x + 1, & x > 1 \end{cases}$, what is $g(x)$ when $x = 0$?

g(x) = 1 Signup and view all the answers

For $g(x) = \begin{cases} 6x + 7, & x \leq -2 \ 4-3, & -2 < x \leq 1 \ -x + 1, & x > 1 \end{cases}$, evaluate $g(x)$ when $x = 2$.

g(x) = -1 Signup and view all the answers

Determine the value of $g(x)$ for $x = -2$ in $g(x) = \begin{cases} 6x + 7, & x \leq -2 \ 4-3, & -2 < x \leq 1 \ -x + 1, & x > 1 \end{cases}$.

g(x) = -5 Signup and view all the answers

Using the function $g(x) = \begin{cases} 6x + 7, & x \leq -2 \ 4-3, & -2 < x \leq 1 \ -x + 1, & x > 1 \end{cases}$, find the value of $g(x)$ at $x = 1.5$.

g(x) = -0.5 Signup and view all the answers

Without finding the inverse, what is the domain and range of the function $f(x) = \sqrt{x+2}$?

Domain: $x \geq -2$, Range: $f(x) \geq 0$ Signup and view all the answers

Without finding the inverse, what is the domain and range of the function $f(x) = \frac{x-1}{x-4}, x \neq 4$?

Domain: $x \neq 4$, Range: $y \neq 1$ Signup and view all the answers

Without finding the inverse, what is the domain and range of the function $f(x) = \frac{1}{x+3}, x \neq -3$?

Domain: $x \neq -3$, Range: $y \neq 0$ Signup and view all the answers

Without finding the inverse, what is the domain and range of the function $f(x) = (x-5)^2, x \geq 5$?

Domain: $x \geq 5$, Range: $f(x) \geq 0$ Signup and view all the answers

What does the symbol $x \to 0$ represent?

It represents that $x$ approaches zero. Signup and view all the answers

What is the difference between $x \to 0$ and $x = 0$?

The symbol $x \to 0$ means $x$ approaches zero, while $x = 0$ means $x$ is exactly zero. Signup and view all the answers

Study Notes

Exercise 1.1

Find the value of f(x) for different inputs:
- f(-2) and f(0) for given functions
- f(x-1) and f(x^2+4) for given functions

Exercise 1.2

Find f(a+h) - f(a) and simplify for different functions:
- f(x) = h/(6x-9)
- f(x) = x^2 + 2x - 1
- f(x) = sin(x)
- f(x) = cos(x)

Exercise 1.3

Express functions in terms of different variables:
- Perimeter P of a square as a function of its area A
- Area A of a circle as a function of its circumference C
- Volume V of a cube as a function of the area A of its base

Composition of Functions

Composition of functions: (f ∘ g)(x) = f(g(x))
Examples of composite functions:
- g(x) = 2x - 5
- g(x) = √(x^2 - 4)
- g(x) = √(x + 1)
- g(x) = |x - 3|

Exercise 1.4

Find the domain and range of functions:
- g(x) = 2x - 5
- g(x) = √(x^2 - 4)
- g(x) = √(x + 1)
- g(x) = |x - 3|

Composition of Functions (continued)

Find the composition of functions:
- (f ∘ g)(x) and (g ∘ f)(x) for different functions
- (f ∘ f)(x) and (g ∘ g)(x) for different functions

Inverse Functions

Find the inverse of functions:
- f(x) = -2x + 8
- f(x) = 3x^3 + 7
- f(x) = (-x + 9)^3
- f(x) = 2x + 1/x - 1, x > 1

Domain and Range of Inverse Functions

State the domain and range of inverse functions:
- f(x) = √(x + 2)
- f(x) = x - 1/(x - 4), x ≠ 4
- f(x) = 1/(x + 3), x ≠ -3
- f(x) = (x - 5)^2, x ≥ 5

Limit of a Function

Definition of a limit of a function:
- The concept of limit of a function is the basis of calculus
- The phrase "x approaches zero" means x is becoming smaller and smaller as n increases
- The symbol x → 0 is different from x = 0

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