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A. The parent graph is translated up 0.5 units.
Use the graph of y = x 2 to describe the graph of the related function y = 0.5x 2. A. The parent graph is translated up 0.5 units. B. The parent graph is compressed horizontally by a factor of 0.5. C. The parent graph is compressed vertically by a factor of 0.5. D. The parent graph is translated down 0.5 units. 5–Minute Check 1
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A. The parent graph is translated up 0.5 units.
Use the graph of y = x 2 to describe the graph of the related function y = 0.5x 2. A. The parent graph is translated up 0.5 units. B. The parent graph is compressed horizontally by a factor of 0.5. C. The parent graph is compressed vertically by a factor of 0.5. D. The parent graph is translated down 0.5 units. 5–Minute Check 1
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A. The parent graph is translated left 3 units and up 4 units.
Use the graph of y = x 2 to describe the graph of the related function y = (x – 4)2 – 3. A. The parent graph is translated left 3 units and up 4 units. B. The parent graph is translated right 3 units and down 4 units. C. The parent graph is translated left 4 units and down 3 units. D. The parent graph is translated right 4 units and down 3 units. 5–Minute Check 2
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A. The parent graph is translated left 3 units and up 4 units.
Use the graph of y = x 2 to describe the graph of the related function y = (x – 4)2 – 3. A. The parent graph is translated left 3 units and up 4 units. B. The parent graph is translated right 3 units and down 4 units. C. The parent graph is translated left 4 units and down 3 units. D. The parent graph is translated right 4 units and down 3 units. 5–Minute Check 2
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A. B. C. D. 5–Minute Check 3
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A. B. C. D. 5–Minute Check 3
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A. f (x) = x; g(x) is f(x) translated left 4 units.
Identify the parent function f (x) if and describe how the graphs of g (x) and f (x) are related. A. f (x) = x; g(x) is f(x) translated left 4 units. B. f(x) = |x|; g(x) is f(x) translated right 4 units. C g(x) is f(x) translated right 4 units. D g(x) is f(x) translated left 4 units. 5–Minute Check 4
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A. f (x) = x; g(x) is f(x) translated left 4 units.
Identify the parent function f (x) if and describe how the graphs of g (x) and f (x) are related. A. f (x) = x; g(x) is f(x) translated left 4 units. B. f(x) = |x|; g(x) is f(x) translated right 4 units. C g(x) is f(x) translated right 4 units. D g(x) is f(x) translated left 4 units. 5–Minute Check 4
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Key Concept 1
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(f + g)(x) = f(x) + g(x) Definition of sum of two functions
Operations with Functions A. Given f (x) = x 2 – 2x, g (x) = 3x – 4, and h (x) = –2x 2 + 1, find the function and domain for (f + g)(x). (f + g)(x) = f(x) + g(x) Definition of sum of two functions = (x 2 – 2x) + (3x – 4) f (x) = x 2 – 2x; g (x) = 3x – 4 = x 2 + x – 4 Simplify. The domain of f and g are both so the domain of (f + g) is Answer: Example 1
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(f + g)(x) = f(x) + g(x) Definition of sum of two functions
Operations with Functions A. Given f (x) = x 2 – 2x, g (x) = 3x – 4, and h (x) = –2x 2 + 1, find the function and domain for (f + g)(x). (f + g)(x) = f(x) + g(x) Definition of sum of two functions = (x 2 – 2x) + (3x – 4) f (x) = x 2 – 2x; g (x) = 3x – 4 = x 2 + x – 4 Simplify. The domain of f and g are both so the domain of (f + g) is Answer: Example 1
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(f – h)(x) = f(x) – h(x) Definition of difference of two functions
Operations with Functions B. Given f (x) = x 2 – 2x, g (x) = 3x – 4, and h (x) = –2x 2 + 1, find the function and domain for (f – h)(x). (f – h)(x) = f(x) – h(x) Definition of difference of two functions = (x 2 – 2x) – (–2x 2 + 1) f(x) = x 2 – 2x; h(x) = –2x 2 + 1 = 3x 2 – 2x – 1 Simplify. The domain of f and h are both so the domain of (f – h) is Answer: Example 1
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(f – h)(x) = f(x) – h(x) Definition of difference of two functions
Operations with Functions B. Given f (x) = x 2 – 2x, g (x) = 3x – 4, and h (x) = –2x 2 + 1, find the function and domain for (f – h)(x). (f – h)(x) = f(x) – h(x) Definition of difference of two functions = (x 2 – 2x) – (–2x 2 + 1) f(x) = x 2 – 2x; h(x) = –2x 2 + 1 = 3x 2 – 2x – 1 Simplify. The domain of f and h are both so the domain of (f – h) is Answer: Example 1
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(f ● g)(x) = f (x) ● g(x) Definition of product of two functions
Operations with Functions C. Given f (x) = x 2 – 2x, g(x) = 3x – 4, and h (x) = –2x 2 + 1, find the function and domain for (f ● g)(x). (f ● g)(x) = f (x) ● g(x) Definition of product of two functions = (x 2 – 2x)(3x – 4) f (x) = x 2 – 2x; g (x) = 3x – 4 = 3x 3 – 10x 2 + 8x Simplify. The domain of f and g are both so the domain of (f ● g) is Answer: Example 1
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(f ● g)(x) = f (x) ● g(x) Definition of product of two functions
Operations with Functions C. Given f (x) = x 2 – 2x, g(x) = 3x – 4, and h (x) = –2x 2 + 1, find the function and domain for (f ● g)(x). (f ● g)(x) = f (x) ● g(x) Definition of product of two functions = (x 2 – 2x)(3x – 4) f (x) = x 2 – 2x; g (x) = 3x – 4 = 3x 3 – 10x 2 + 8x Simplify. The domain of f and g are both so the domain of (f ● g) is Answer: Example 1
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Definition of quotient of two functions
Operations with Functions D. Given f (x) = x 2 – 2x, g (x) = 3x – 4, and h (x) = –2x 2 + 1, find the function and domain for Definition of quotient of two functions f(x) = x 2 – 2x; h(x) = –2x 2 + 1 Example 1
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Operations with Functions
The domains of h and f are both (–∞, ∞), but x = 0 or x = 2 yields a zero in the denominator of So, the domain of (–∞, 0) È (0, 2) È (2, ∞). Answer: Example 1
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Answer: D = (–∞, 0) È (0, 2) È (2, ∞)
Operations with Functions The domains of h and f are both (–∞, ∞), but x = 0 or x = 2 yields a zero in the denominator of So, the domain of (–∞, 0) È (0, 2) È (2, ∞). Answer: D = (–∞, 0) È (0, 2) È (2, ∞) Example 1
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Find (f + g)(x), (f – g)(x), (f ● g)(x), and for f (x) = x 2 + x, g (x) = x – 3. State the domain of each new function. Example 1
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A. B. C. D. Example 1
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A. B. C. D. Example 1
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Key Concept 2
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A. Given f (x) = 2x2 – 1 and g (x) = x + 3, find [f ○ g](x).
Compose Two Functions A. Given f (x) = 2x2 – 1 and g (x) = x + 3, find [f ○ g](x). Replace g (x) with x + 3 = f (x + 3) Substitute x + 3 for x in f (x). = 2(x + 3)2 – 1 Expand (x +3)2 = 2(x 2 + 6x + 9) – 1 Simplify. = 2x x + 17 Answer: Example 2
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A. Given f (x) = 2x2 – 1 and g (x) = x + 3, find [f ○ g](x).
Compose Two Functions A. Given f (x) = 2x2 – 1 and g (x) = x + 3, find [f ○ g](x). Replace g (x) with x + 3 = f (x + 3) Substitute x + 3 for x in f (x). = 2(x + 3)2 – 1 Expand (x +3)2 = 2(x 2 + 6x + 9) – 1 Simplify. = 2x x + 17 Answer: [f ○ g](x) = 2x x + 17 Example 2
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B. Given f (x) = 2x2 – 1 and g (x) = x + 3, find [g ○ f](x).
Compose Two Functions B. Given f (x) = 2x2 – 1 and g (x) = x + 3, find [g ○ f](x). Substitute 2x 2 – 1 for x in g (x). = (2x 2 – 1) + 3 Simplify = 2x 2 + 2 Answer: Example 2
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B. Given f (x) = 2x2 – 1 and g (x) = x + 3, find [g ○ f](x).
Compose Two Functions B. Given f (x) = 2x2 – 1 and g (x) = x + 3, find [g ○ f](x). Substitute 2x 2 – 1 for x in g (x). = (2x 2 – 1) + 3 Simplify = 2x 2 + 2 Answer: [g ○ f](x) = 2x 2 + 2 Example 2
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C. Given f (x) = 2x 2 – 1 and g (x) = x + 3, find [f ○ g](2).
Compose Two Functions C. Given f (x) = 2x 2 – 1 and g (x) = x + 3, find [f ○ g](2). Evaluate the expression you wrote in part A for x = 2. [f ○ g](2) = 2(2)2 + 12(2) + 17 Substitute 2 for x. = 49 Simplify. Answer: Example 2
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C. Given f (x) = 2x 2 – 1 and g (x) = x + 3, find [f ○ g](2).
Compose Two Functions C. Given f (x) = 2x 2 – 1 and g (x) = x + 3, find [f ○ g](2). Evaluate the expression you wrote in part A for x = 2. [f ○ g](2) = 2(2)2 + 12(2) + 17 Substitute 2 for x. = 49 Simplify. Answer: [f ○ g](2) = 49 Example 2
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Find for f (x) = 2x – 3 and g (x) = 4 + x 2.
A. 2x ; 4x 2 – 12x + 13; 23 B. 2x ; 4x 2 – 12x + 5; 23 C. 2x 2 + 5; 4x 2 – 12x + 5; 23 D. 2x 2 + 5; 4x 2 – 12x + 13; 23 Example 2
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Find for f (x) = 2x – 3 and g (x) = 4 + x 2.
A. 2x ; 4x 2 – 12x + 13; 23 B. 2x ; 4x 2 – 12x + 5; 23 C. 2x 2 + 5; 4x 2 – 12x + 5; 23 D. 2x 2 + 5; 4x 2 – 12x + 13; 23 Example 2
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Find a Composite Function with a Restricted Domain
A. Find Example 3
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Find a Composite Function with a Restricted Domain
To find , you must first be able to find g(x) = (x – 1) 2, which can be done for all real numbers. Then you must be able to evaluate for each of these g (x)-values, which can only be done when g (x) > 1. Excluding from the domain those values for which 0 < (x – 1) 2 <1, namely when 0 < x < 1, the domain of f ○ g is (–∞, 0] È [2, ∞). Now find [f ○ g](x). Example 3
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Substitute (x – 1)2 for x in f (x).
Find a Composite Function with a Restricted Domain Replace g (x) with (x – 1)2. Substitute (x – 1)2 for x in f (x). Simplify. Notice that is not defined for 0 < x < 2. Because the implied domain is the same as the domain determined by considering the domains of f and g, we can write the composition as for (–∞, 0] È [2, ∞). Example 3
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Find a Composite Function with a Restricted Domain
Answer: Example 3
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Find a Composite Function with a Restricted Domain
Answer: for (–∞, 0] È [2, ∞). Example 3
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B. Find f ○ g. Find a Composite Function with a Restricted Domain
Example 3
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Find a Composite Function with a Restricted Domain
To find f ○ g, you must first be able to find , which can be done for all real numbers x such that x2 1. Then you must be able to evaluate for each of these g (x)-values, which can only be done when g (x) 0. Excluding from the domain those values for which 0 > x 2 – 1, namely when –1 < x < 1, the domain of f ○ g is (–∞, –1) È (1, ∞). Now find [f ○ g](x). Example 3
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Find a Composite Function with a Restricted Domain
Example 3
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Find a Composite Function with a Restricted Domain
Answer: Example 3
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Find a Composite Function with a Restricted Domain
Answer: Example 3
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Find f ○ g. A. D = (–∞, –1) (–1, 1) (1, ∞); B. D = [–1, 1];
C. D = (–∞, –1) (–1, 1) (1, ∞); D. D = (0, 1); Example 3
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Find f ○ g. A. D = (–∞, –1) (–1, 1) (1, ∞); B. D = [–1, 1];
C. D = (–∞, –1) (–1, 1) (1, ∞); D. D = (0, 1); Example 3
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B. Find two functions f and g such that
Decompose a Composite Function B. Find two functions f and g such that when h (x) = 3x 2 – 12x Neither function may be the identity function f (x) = x. h (x) = 3x2 – 12x + 12 Notice that h is factorable. = 3(x2 – 4x + 4) or 3(x – 2) 2 Factor. One way to write h (x) as a composition is to let f (x) = 3x2 and g (x) = x – 2. Example 4
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Decompose a Composite Function
Sample answer: Example 4
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Sample answer: g (x) = x – 2 and f (x) = 3x 2
Decompose a Composite Function Sample answer: g (x) = x – 2 and f (x) = 3x 2 Example 4
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A. B. C. D. Example 4
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A. B. C. D. Example 4
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So, the functions are R(t) = 25 + 10t and A(R) = R 2.
Compose Real-World Functions A. COMPUTER ANIMATION An animator starts with an image of a circle with a radius of 25 pixels. The animator then increases the radius by 10 pixels per second. Find functions to model the data. The length r of the radius increases at a rate of 10 pixels per second, so R(t) = t, where t is the time in seconds and t 0. The area of the circle is times the square of the radius. So, the area of the circle is A(R) = R 2. So, the functions are R(t) = t and A(R) = R 2. Answer: Example 5
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So, the functions are R(t) = 25 + 10t and A(R) = R 2.
Compose Real-World Functions A. COMPUTER ANIMATION An animator starts with an image of a circle with a radius of 25 pixels. The animator then increases the radius by 10 pixels per second. Find functions to model the data. The length r of the radius increases at a rate of 10 pixels per second, so R(t) = t, where t is the time in seconds and t 0. The area of the circle is times the square of the radius. So, the area of the circle is A(R) = R 2. So, the functions are R(t) = t and A(R) = R 2. Answer: R(t) = t; A(R) = R 2 Example 5
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A ○ R = A[R(t)] Definition of A ○ R
Compose Real-World Functions B. COMPUTER ANIMATION An animator starts with an image of a circle with a radius of 25 pixels. The animator then increases the radius by 10 pixels per second. Find A ○ R. What does the function represent? A ○ R = A[R(t)] Definition of A ○ R =A( t) Replace R (t) with t. = ( t)2 Substitute ( t) for R in A(R). = 100t t + 625 Simplify. Example 5
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Compose Real-World Functions
So, A ○ R = 100t t + 625. The composite function models the area of the circle as a function of time. Answer: Example 5
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Compose Real-World Functions
So, A ○ R = 100t t + 625. The composite function models the area of the circle as a function of time. Answer: A ○ R = 100t t + 625 ; the function models the area of the circle as a function of time. Example 5
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