What are the domain and range of $$ f(x)=4^{x}-8 $$ ? domain: $$ \{x \mid x $$ is a real number $$ \} $$; range: $$ \{y \mid y-8\} $$ domain: $$ \{x \mid x $$ is a real number $$ \} $$; range: $$ \{y \mid y8\} $$ domain: $$ \{x \mid x4\} $$; range: $$ \{y \mid y-8\} $$ domain: $$ \{x \mid x-4\} $$; range: $$ \{y \mid y8\} $$

Question

What are the domain and range of $$ f(x)=4^{x}-8 $$ ? domain: $$ \{x \mid x $$ is a real number $$ \} $$; range: $$ \{y \mid y>-8\} $$ domain: $$ \{x \mid x $$ is a real number $$ \} $$; range: $$ \{y \mid y>8\} $$ domain: $$ \{x \mid x>4\} $$; range: $$ \{y \mid y>-8\} $$ domain: $$ \{x \mid x>-4\} $$; range: $$ \{y \mid y>8\} $$

UpStudy AI · Accepted Answer

**Step 1: Determine the Domain**

The function is given by 
$$
f(x) = 4^x - 8.
$$
Since the term $$4^x$$ is defined for every real number $$x$$ (as an exponential function), the domain of $$f(x)$$ is all real numbers. In set notation, this is:
$$
\{\, x \mid x \text{ is a real number} \,\}.
$$

**Step 2: Determine the Range**

The function $$4^x$$ takes values in $$(0, \infty)$$, because an exponential function with a positive base is always positive. Thus,
$$
4^x \in (0,\infty).
$$
Subtracting 8 from $$4^x$$, we obtain:
$$
f(x) = 4^x - 8.
$$
This shifts the entire graph of $$4^x$$ downward by 8 units. Therefore, the output values of $$f(x)$$ will be all values greater than $$-8$$. Mathematically, the range is:
$$
\{\, y \mid y > -8 \,\}.
$$

**Correct Answer**

domain: $$ \{x \mid x \text{ is a real number}\} $$; range: $$ \{y \mid y > -8\} $$.
The domain is all real numbers, and the range is all $$ y $$ such that $$ y > -8 $$.

What are the domain and range of ?
domain: is a real number ; range:
domain: is a real number ; range:
domain: ; range:
domain: ; range:

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