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Problem 1

# For each function, evaluate (a) $$f(0,0)$$; (b) $$f(1,0) ;$$ (c) $$f(0,-1)$$; (d) $$f(a, 2) ;$$ (e) $$f(y, x)$$;(f) $$f(x+h, y+k)$$ HINT [See Quick Examples page 1080.] $$f(x, y)=x^{2}+y^{2}-x+1$$

### Short Answer

Expert verified
The short answers for the given function, $$f(x,y) = x^2 + y^2 - x + 1$$, evaluated at different inputs are: (a) $$f(0,0) = 1$$ (b) $$f(1,0) = 1$$ (c) $$f(0,-1) = 2$$ (d) $$f(a,2) = a^2 - a + 5$$ (e) $$f(y,x) = y^2 + x^2 - y + 1$$ (f) $$f(x+h, y+k) = x^2 + 2xh + h^2 + y^2 + 2yk + k^2 - x - h + 1$$
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## Step 1: (a)

: To find $$f(0,0)$$, we plug in $$x = 0$$ and $$y = 0$$ into the function: $$f(0, 0) = (0)^2 + (0)^2 - (0) + 1 = 0 + 0 + 1 = 1$$

## Step 2: (b)

: To find $$f(1,0)$$, we plug in $$x = 1$$ and $$y = 0$$ into the function: $$f(1, 0) = (1)^2 + (0)^2 - (1) + 1 = 1 + 0 - 1 + 1 = 1$$

## Step 3: (c)

: To find $$f(0,-1)$$, we plug in $$x = 0$$ and $$y = -1$$ into the function: $$f(0, -1) = (0)^2 + (-1)^2 - (0) + 1 = 0 + 1 + 1 = 2$$

## Step 4: (d)

: To find $$f(a, 2)$$, we plug in $$x = a$$ and $$y = 2$$ into the function: $$f(a, 2) = a^2 + (2)^2 - a + 1 = a^2 + 4 - a + 1 = a^2 - a + 5$$

## Step 5: (e)

: To find $$f(y, x)$$, we plug in $$x = y$$ and $$y = x$$ into the function: $$f(y, x) = y^2 + x^2 - y + 1$$

## Step 6: (f)

: To find $$f(x+h, y+k)$$, we plug in $$x = x + h$$ and $$y = y + k$$ into the function: $$f(x+h, y+k) = (x + h)^2 + (y + k)^2 - (x + h) + 1 = x^2 + 2xh + h^2 + y^2 + 2yk + k^2 - x - h + 1$$

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