Ques 1)
We know that the transformation of the type:
f(x) → f(x-k)
shifts the graph of the parent function f(x) k units to the right.
Hence, on replacing x with x−4 the graph of the function f(x) will be shifted 4 units to the right.
The correct option is:
The graph is shifted 4 units right.
Ques 2)
For finding the hole of a rational function we first factor the numerator and denominator of the function and then the roots of the factor which is common to both numerator and denominator is the hole of the graph.
Here we have a rational function as:
[tex]f(x)=\dfracPx^2+3x-28}{x+7}[/tex]
which on factoring we get:
[tex]f(x)=\dfrac{(x+7)(x-4)}{x+7}[/tex]
Hence, the common factor is: [tex]x+7[/tex]
and the root of the factor is: x= -7
Hence, the hole in the graph of function f(x) is:
x= -7
Ques 3)
We are given a function h(x) as:
h(x)= {x if x≤ -1
and -x if x> -1
The graph is a straight increasing line with positive slope in the interval (-∞,-1] and there is a closed circle at (-1,-1)
and also the graph is a straight decreasing line with negative slope in the interval (-1,∞) and there is a open circle at (-1,1).
The correct graph is attached to the answer.
A function assigns the value of each element of one set to the other specific element of another set. The graph is shifted to 4 units right.
A function assigns the value of each element of one set to the other specific element of another set.
A.) As the graph is transforming from f(x) to f(x-k), therefore, it will shift the graph of the parent function f(x) by k units, towards the right.
Now, since the function is transforming from f(x) to f(x-4), thus, the function of the graph will be shifting 4 units to the right.
Hence, the correct option is The graph is shifted to 4 units right.
B.) For finding the hole of a rational function, we need to find the common factor between the denominator and the numerator of the function.
[tex]f(x)=\dfrac{x^2+3x-28}{x+7}\\\\f(x)=\dfrac{x^2+7x-4x-28}{x+7}\\\\f(x)=\dfrac{x(x+7)-4(x+7)}{x+7}\\\\f(x)=\dfrac{(x+7)(x-4)}{x+7}[/tex]
Since the common factor between the numerator and denominator is (x+7), therefore, equating the factor with 0. we will get,
[tex]x+7=0\\\\x=-7[/tex]
Hence, the coordinates of the hole in the graph of the function f(x) are at -7.
C.) We are given a function h(x) as:
h(x)= {x if x≤ -1,
-x if x> -1
As can be seen, the graph is a straight ascending line with a positive slope in the interval (-∞,-1], with a closed circle at the intersection (-1,-1).
In addition, the graph is a straight descending line with a negative slope in the interval (-1,∞), with an open circle at the intersection (-1,1).
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