Answer:
The domain is (-∞,∞) and range is (k,∞)
Step-by-step explanation:
The graph is given as [tex]a^(x+h)+k[/tex].
When x =0, we have f(x) [tex]a^h+k[/tex]= y intercept
Consider the exponential function y = [tex]a^x[/tex].
This is defined for domain all real numbers and range = (0,infinity) for all positive a.
This function is transformed into our function by horizontal shift of h units to left and vertical units of k units up.
i.e. Y = y-k and X = x-h
Since x has domain as set of all real numbers X also set of real numbers
So domain is (-∞,∞)
Since range of original graph is (0,∞), we have Y minimum value as k and max as infinity.
Range = (k,∞)
The correct option is [tex]\boxed{\bf option (D)}[/tex] i.e., domian is [tex]\boxed{\bf (-\infty,\infty)}[/tex] and the range is [tex]\boxed{(k,\infty)}[/tex].
Further explanation:
It is given that the function is [tex]f(x)=a^{(x+h)}+k[/tex].
First we consider the exponential function [tex]f(x)=a^{x}[/tex], the domain of this function is [tex](-\infty,\infty)[/tex] and range is [tex](0,\infty)[/tex].
The domain of a function is defined as all possible value of [tex]x[/tex] which satisfy the function.
The range of a function is defined as the all possible outcome of the function that is the possible values of [tex]y[/tex].
The given graph of the function [tex]f(x)=a^{(x+h)}+k[/tex] is the transformation of the graph of the function [tex]f(x)=a^{x}[/tex].
If a constant is added to the argument of the function, the graph of the function shifts to the left if the constant is positive i.e., [tex]f(x+a)[/tex] and it shifts to the right if the constant is negative i.e., [tex]f(x-a)[/tex].
If a constant is added to a function, the graph of the function shifts vertically upwards if the constant is positive i.e., [tex]f(x)+a[/tex] and it shifts vertically downwards if the constant is negative i.e., [tex]f(x)-a[/tex].
In the given function constant [tex]h[/tex] is added to the argument of the function [tex]f(x)=a^{x}[/tex], so the graph of the function shifts to the left.
And also in the given function constant [tex]k[/tex] is added to the function [tex]f(x)=a^{x}[/tex], so the graph of the function shifts vertically upwards.
Therefore, the domain of this transformed function [tex]f(x)=a^{(x+h)}+k[/tex] will remain the same [tex](-\infty,\infty)[/tex] as the original function [tex]f(x)=a^{x}[/tex] as all the value of [tex]x[/tex] satisfy this transformed function and the range of the function will become [tex](k,\infty)[/tex] as the minimum value of [tex]y[/tex] is [tex]k[/tex] and maximum is [tex]\infty[/tex].
Now we will check from the given option that which option is correct step by step.
Option (A):
In option (A) the domain is given as [tex](h,\infty)[/tex] and the range is [tex](-\infty,\infty)[/tex] which is not matching with the above answer, so the option (A) is incorrect.
Option (B):
In option (B) the domain is given as [tex](-\infty,\infty)[/tex] and the range is [tex](h,\infty)[/tex] whichis not matching with the above answer, so the option (B) is also incorrect.
Option (C):
In option (C) the domain is given as [tex](h,\infty)[/tex] and the range is [tex](k,\infty)[/tex] whichis not matching with the above answer, so the option (C) is also incorrect.
Option (D):
In option (D) the domain is given as [tex](-\infty,\infty)[/tex] and the range is [tex](k,\infty)[/tex] which is matching with the above answer, so the option (D) is correct.
Learn more
1. Problem on coordinates of triangles after rotation https://brainly.com/question/7437053
2. Problem on transformation of the triangle when rotated about origin https://brainly.com/question/2992432
Answer details:
Grade: High school
Subject: Mathematics
Topic: Shifting of Curve
Keywords: Function, range, domain, f(x)=a^(x+h)+k, constant, vertically, f(x)=a^(x), argument, graph, left, exponential, maximum, argument, shifting, translation, transformation, exponential function.
