Question

If this is the graph of f(x) = a^(x+h) + k then

Answer 1

Answer:

The domain is (-∞,∞) and range is (k,∞)

Step-by-step explanation:

The graph is given as $a^(x+h)+k$.

When x =0, we have f(x) $a^h+k$= y intercept

Consider the exponential function y = $a^x$.

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,∞)

Answer 2

The correct option is $\boxed{\bf option (D)}$ i.e., domian is $\boxed{\bf (-\infty,\infty)}$ and the range is $\boxed{(k,\infty)}$.

Further explanation:

It is given that the function is $f(x)=a^{(x+h)}+k$.

First we consider the exponential function $f(x)=a^{x}$, the domain of this function is $(-\infty,\infty)$ and range is $(0,\infty)$.

The domain of a function is defined as all possible value of $x$ 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 $y$.

The given graph of the function $f(x)=a^{(x+h)}+k$ is the transformation of the graph of the function $f(x)=a^{x}$.

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., $f(x+a)$ and it shifts to the right if the constant is negative i.e., $f(x-a)$.

If a constant is added to a function, the graph of the function shifts vertically upwards if the constant is positive i.e., $f(x)+a$ and it shifts vertically downwards if the constant is negative i.e., $f(x)-a$.

In the given function constant $h$ is added to the argument of the function $f(x)=a^{x}$, so the graph of the function shifts to the left.

And also in the given function constant $k$ is added to the function $f(x)=a^{x}$, so the graph of the function shifts vertically upwards.

Therefore, the domain of this transformed function $f(x)=a^{(x+h)}+k$ will remain the same $(-\infty,\infty)$ as the original function $f(x)=a^{x}$ as all the value of $x$ satisfy this transformed function and the range of the function will become $(k,\infty)$ as the minimum value of $y$ is $k$ and maximum is $\infty$.

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 $(h,\infty)$ and the range is $(-\infty,\infty)$ which is not matching with the above answer, so the option (A) is incorrect.

Option (B):

In option (B) the domain is given as $(-\infty,\infty)$ and the range is $(h,\infty)$ whichis not matching with the above answer, so the option (B) is also incorrect.

Option (C):

In option (C) the domain is given as $(h,\infty)$ and the range is $(k,\infty)$ whichis not matching with the above answer, so the option (C) is also incorrect.

Option (D):

In option (D) the domain is given as $(-\infty,\infty)$ and the range is $(k,\infty)$ which is matching with the above answer, so the option (D) is correct.

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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.